UNIVERSIDAD NACIONAL AUTÓNOMA DE MÉXICO
POSGRADO EN CIENCIAS FÍSICAS
CLASSICAL AND QUANTUM DESCRIPTIONS OF THE
PARAMETER SPACE GEOMETRY
TESIS
QUE PARA OPTAR POR EL GRADO DE:
DOCTOR EN CIENCIAS (FÍSICA)
PRESENTA:
DANIEL GUTIÉRREZ RUIZ
TUTOR PRINCIPAL
DR. JOSÉ DAVID VERGARA OLIVER
Instituto de Ciencias Nucleares
COMITÉ TUTOR
DR. HERNANDO QUEVEDO CUBILLOS
Instituto de Ciencias Nucleares
DRA. GABRIELA MURGUÍA ROMERO
Facultad de Ciencias
CIUDAD UNIVERSITARIA, CD. MX., MARZO DE 2021
 
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Contents
Abstract vii
Resumen ix
Acknowledgments xi
List of publications xiii
Introduction xv
1. Quantum phase transitions and geometry 1
1.1 Fidelity approach to quantum phase transitions . . . . . . . . . . . . . . 1
1.2 Elements of Riemannian geometry . . . . . . . . . . . . . . . . . . . . . . 3
2. Quantum geometric tensor 7
2.1 The Berry phase, connection and curvature . . . . . . . . . . . . . . . . . 7
2.2 Quantum metric tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Abelian and non-Abelian quantum geometric tensor . . . . . . . . . . . . 10
2.4 Generator approach to the quantum geometric tensor . . . . . . . . . . . 12
2.5 Path integral approach to the quantum geometric tensor . . . . . . . . . 13
2.6 Equivalence between the two approaches . . . . . . . . . . . . . . . . . . 15
3. Phase space formulation of the quantum geometric tensor 19
3.1 Wigner function formalism . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Phase space formulation of the parameter space geometry . . . . . . . . . 20
3.2.1 Abelian case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.2 Non-Abelian case . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 Illustrative examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3.1 Generalized harmonic oscillator . . . . . . . . . . . . . . . . . . . 26
3.3.2 Three coupled harmonic oscillators . . . . . . . . . . . . . . . . . 28
4. Classical analog of the quantum geometric tensor 31
4.1 The Hannay angle, connection and curvature . . . . . . . . . . . . . . . . 31
4.2 Classical analog of the quantum metric tensor . . . . . . . . . . . . . . . 35
4.2.1 Behavior under canonical transformations . . . . . . . . . . . . . 36
iii
iv Contents
4.3 Time-dependent approach to the classical metric and curvature . . . . . . 38
4.4 Equivalence between the two approaches . . . . . . . . . . . . . . . . . . 41
4.4.1 Classical metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.4.2 The Hannay curvature . . . . . . . . . . . . . . . . . . . . . . . . 46
5. Examples of the generator approach to the classical metric 49
5.1 Generalized harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . 49
5.1.1 Standard generalized harmonic oscillator . . . . . . . . . . . . . . 49
5.1.2 Linear term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.2 Quartic anharmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . 54
5.3 Two-dimensional isotropic harmonic oscillator in polar coordinates . . . . 60
5.4 Two-dimensional attractive Coulomb problem . . . . . . . . . . . . . . . 65
5.5 Two-dimensional anisotropic harmonic oscillator . . . . . . . . . . . . . . 68
5.6 Mapping of metrics through the Kustaanheimo-Stiefel transformation . . 69
6. Examples of the time-dependent approach to the classical metric 73
6.1 Symmetric coupled harmonic oscillators . . . . . . . . . . . . . . . . . . . 73
6.2 Linearly coupled harmonic oscillators . . . . . . . . . . . . . . . . . . . . 78
6.3 Singular Euclidean oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.4 Spin-half particle in an external magnetic field . . . . . . . . . . . . . . . 92
7. Application to the Dicke model 99
7.1 Analysis in the thermodynamic limit . . . . . . . . . . . . . . . . . . . . 100
7.1.1 Normal phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.1.2 Superradiant phase . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.2 Classical and quantum metric tensors . . . . . . . . . . . . . . . . . . . . 102
7.2.1 Normal phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7.2.2 Superradiant phase . . . . . . . . . . . . . . . . . . . . . . . . . . 104
7.2.3 Metrics under resonance . . . . . . . . . . . . . . . . . . . . . . . 105
8. Application to the anisotropic Lipkin-Meshkov-Glick model 109
8.1 Analysis in the thermodynamic limit . . . . . . . . . . . . . . . . . . . . 110
8.1.1 Symmetric phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
8.1.2 Broken phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
8.2 Classical and quantum metric tensors . . . . . . . . . . . . . . . . . . . . 112
8.2.1 Metrics for the symmetric phase . . . . . . . . . . . . . . . . . . . 113
8.2.2 Metrics for the broken phase . . . . . . . . . . . . . . . . . . . . . 115
8.2.3 Finite-size analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 117
9. Application to a modified Lipkin-Meshkov-Glick model 127
9.1 Analysis in the thermodynamic limit . . . . . . . . . . . . . . . . . . . . 127
9.2 Quantum geometric tensor for the LMG model . . . . . . . . . . . . . . . 135
9.2.1 Ground state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Contents v
9.2.2 Highest energy state . . . . . . . . . . . . . . . . . . . . . . . . . 140
9.2.2.1 Symmetric phase . . . . . . . . . . . . . . . . . . . . . . 140
9.2.2.2 Broken phase . . . . . . . . . . . . . . . . . . . . . . . . 141
9.2.3 Analysis of the peaks . . . . . . . . . . . . . . . . . . . . . . . . . 148
9.2.3.1 Case 1: ξy = 2.3 . . . . . . . . . . . . . . . . . . . . . . 148
9.2.3.2 Case 2: Ωx = 0 . . . . . . . . . . . . . . . . . . . . . . . 149
9.2.3.3 Case 3: ξy = 0.5 . . . . . . . . . . . . . . . . . . . . . . 149
10.Relativistic Coulomb problem from N = 4 super Yang-Mills 153
10.1 Modified Klein-Gordon equation from N = 4 SYM . . . . . . . . . . . . 153
10.2 Modified relativistic particle . . . . . . . . . . . . . . . . . . . . . . . . . 157
10.3 Relativistic SO(4) algebra using the relativistic Runge-Lenz vector . . . . 159
10.4 Relativistic Kustaanheimo-Stiefel duality . . . . . . . . . . . . . . . . . . 162
11.Conclusions 167
Appendices 171
A. Classical generators for the quartic anharmonic oscillator 171
B. Grassmann variables 173
C. Bloch coherent states 175
D. Wolfram Mathematica codes 179
D.1 Quantum metric tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
D.2 Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
D.3 Scalar curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
E. Quantum geometric tensor for Bloch coherent states in the modified
LMG model 183
F. Precision considerations in the modified LMG model 187
References 200

Abstract
This thesis deals with the geometrical structures that underlie the parameter space of a
physical system. We begin by introducing the Berry phase, connection and curvature.
The Berry phase is the flux of the Berry curvature and emerges in a quantum system when
we traverse adiabatically a circuit in parameter space. After understanding this object,
we present the quantum metric tensor which encodes how distances are measured in
parameter space in a gauge-invariant manner. It turns out that both the Berry curvature
and the quantum metric tensor can be obtained, respectively, as the imaginary and real
part of the quantum geometric tensor. The description can be broadened to include
also a non-Abelian gauge structure that appears when considering degenerate quantum
states. All these results are then cast in terms of Hermitian operators that generate
displacements in parameter space, which will later turn out to be of special importance.
Later, we introduce a path integral approach to treat the quantum geometric tensor and
prove the equivalence between this description and the initial perspective based on the
Hilbert space.
Once the basic tools have been presented, we introduce a novel phase space formu-
lation of the parameter space geometry whose protagonist is the Wigner function. We
establish the Abelian and non-Abelian descriptions and illustrate the approach with two
examples.
In the following part, with the help of the classical analogs of the generators of
displacements in parameter space, we reformulate the Hannay curvature, which is the
classical analog of the Berry curvature. After this, we reach the central part of the work
by introducing the classical analog of the quantum metric tensor, a novel object that
measures the distance in the parameter space of a classical integrable system. Then,
we present an equivalent time-dependent approach that arises from the semiclassical
approximation of the path integral approach to the quantum geometric tensor.
We devote a good part of this thesis to the presentation of examples that illustrate
the use of the different formulations and help us establish a comparison between the
quantum and classical results. This will form the basis for the analysis of the Dicke and
Lipkin-Meshkov-Glick models, where we use the quantum and classical metric tensors
and their scalar curvatures to study quantum phase transitions in the thermodynamic
limit, as well as their precursors for finite sizes of the system.
Finally, we find a relativistic Runge-Lenz vector for a modified Klein-Gordon equation
that preserves the SO(4) symmetry. We describe its extraction from the N = 4 super
vii
viii Abstract
Yang-Mills theory through a particular Higgs mechanism and connect its symmetries
with those of the relativistic harmonic oscillator.
Resumen
Esta tesis trata sobre las estructuras geométricas subyacentes al espacio de parámetros
de un sistema físico. Comenzamos presentando la fase de Berry, su conexión y curvatura.
La fase de Berry es el flujo de la curvatura de Berry y aparece en un sistema cuántico
cuando recorremos adiabáticamente un circuito en el espacio de parámetros. Después
de entender este objeto, introducimos el tensor métrico cuántico, el cual codifica cómo
se miden las distancias en el espacio de parámetros en una forma invariante de norma.
Vemos que tanto la curvatura de Berry como el tensor métrico cuántico pueden obten-
erse, respectivamente, como la parte imaginaria y real del tensor geométrico cuántico.
Esta descripción puede ampliarse para incluir una estructura de norma no Abeliana que
aparece al considerar estados cuánticos degenerados. Todos estos resultados se formulan
luego en términos de operadores Hermitianos que generan desplazamientos en el espacio
de parámetros, que serán luego de especial importancia. Posteriormente, introducimos
el enfoque de integral de trayectoria del tensor geométrico cuántico y demostramos la
equivalencia entre esta descripción y el enfoque inicial basado en el espacio de Hilbert.
Ya que las herramientas básicas fueron introducidas, presentamos una nueva formu-
lación en el espacio fase de la geometría del espacio de parámetros cuyo protagonista
es la función de Wigner. Establecemos las descripciones Abelianas y no Abelianas e
ilustramos el enfoque con dos ejemplos.
En la siguiente parte, mediante los análogos clásicos de los generadores de desplaza-
mientos en el espacio de parámetros, reformulamos la curvatura de Hannay, que es el
análogo clásico de la curvatura de Berry. Después, llegamos a la parte central del trabajo
al introducir el análogo clásico del tensor métrico cuántico, un objeto nuevo que mide la
distancia en el espacio de parámetros de un sistema integrable clásico. Luego, presenta-
mos una fomulación equivalente dependiente del tiempo y que surge de la aproximación
semiclásica del enfoque de integral de trayectoria del tensor geométrico cuántico.
Dedicamos una buena parte de esta tesis a presentar ejemplos que ilustren el uso
de las diferentes formulaciones y nos ayuden a establecer una comparación entre los
resultados cuánticos y clásicos. Esto formará la base para el análisis de los modelos
de Dicke y de Lipkin-Meshkov-Glick, donde usamos el tensor métrico cuántico y clásico
y sus curvaturas escalares para estudiar las transiciones de fase cuánticas en el límite
termodinámico, así como sus precursores para tamaños finitos del sistema.
Finalmente, encontramos un vector relativista de Runge-Lenz para una ecuación de
Klein-Gordon modificada que preserva la simetría SO(4). Describimos su obtención
ix
x Resumen
a partir de la teoría super Yang-Mills N = 4 mediante cierto mecanismo de Higgs y
conectamos sus simetrías con las del oscilador armónico relativista.
Acknowledgments
I wish to thank:
• God for filling me with his grace and blessing me with the opportunity to get
through my studies.
• Dr. José David Vergara for being an admirable supervisor. Your valuable help,
patience and comprehension allowed me to finish this work.
• Drs. Hernando Quevedo and Gabriela Murguía for their advice and always being
attentive to my progress.
• Drs. Jorge Hirsch, Andrea Valdés, and David José Fernández for their suggestions
and comments to improve this work.
• Dr. Diego González for his support and patience. Without your help and ideas, I
am sure that this thesis would be half its length.
• Drs. Jorge Hirsch, Antonio García, and Jorge Chávez for wonderful collaborations
together.
• Dr. Eduardo Rosas for his wise advice that was essential during this long journey.
I am truly indebted to you.
• My family for their unconditional love and support. This thesis is dedicated to
you, mom.
• Ema, the most wonderful person in the universe. I could write two hundred more
pages telling how much you mean to me.
• All my friends and partners for always putting a smile on my face.
• The staff of the ICN and the PCF for their help with administrative matters.
• CONACYT for supporting me with the four-year scholarship No. 463765.
• The DGAPA-PAPIIT Grant No. IN103919 provided by UNAM.
xi

List of publications
The results of this thesis have been published on the following peer-reviewed articles:
• J. Alvarez-Jimenez, I. Cortese, J. Antonio García, D. Gutiérrez-Ruiz, and J. David
Vergara, “Relativistic Runge-Lenz vector: from N = 4 SYM to SO(4) scalar field
theory”, Journal of High Energy Physics 2018, 153 (2018)
• Diego Gonzalez, D. Gutiérrez-Ruiz, and J. David Vergara, “Classical analog of the
quantum metric tensor”, Physical Review E 99, 032144 (2019)
• J. Alvarez-Jimenez, Diego Gonzalez, D. Gutiérrez-Ruiz, and J. David Vergara,
“Geometry of the parameter space of a quantum system: Classical point of view”,
Annalen der Physik 532, 1900215 (2020)
• Diego Gonzalez, D. Gutiérrez-Ruiz, and J. David Vergara, “Phase space formula-
tion of the Abelian and non-Abelian quantum geometric tensor”, Journal of Physics
A: Mathematical and Theoretical 53, 505305 (2020)
and on the conference article:
• J. Alvarez-Jimenez, I. Cortese, J. Antonio García, D. Gutiérrez-Ruiz, and J. David
Vergara, “Reestablishment of SO(4) symmetry in the relativistic Coulomb problem
from N = 4 SYM”, Journal of Physics: Conference Series 1194, 012057 (2019).
Additional peer-reviewed collaborations:
• J. Antonio García, D. Gutiérrez-Ruiz, and R. Abraham Sánchez-Isidro, “Vertical
extension of Noether theorem for scaling symmetries”, European Physical Journal
Plus 136, 3 (2021).
• D. Gutiérrez-Ruiz, Diego Gonzalez, Jorge Chávez-Carlos, Jorge G. Hirsch, and J.
David Vergara, “Quantum geometric tensor and quantum phase transitions in the
Lipkin-Meshkov-Glick model”, Physical Review B (sent).
• Diego Gonzalez, D. Gutiérrez-Ruiz, and J. David Vergara, “Classical and quantum
descriptions of the parameter space geometry in the Dicke and Lipkin-Meshkov-
Glick models”, Physical Review E (sent).
xiii

Introduction
There is a deep interplay between geometry, information theory and quantum mechanics
[1]. A fundamental element in geometry is the metric tensor which allows to measure
distances. A metric of great interest that is defined for statistical spaces is the Fisher
metric. This metric plays an important role in the study of information in quantum
mechanics [2], where it is called Bures metric [3] for mixed states, and Fubini-Study
metric, or quantum geometric tensor for pure states [4]. When the real part of this
tensor is taken, the quantum metric tensor is obtained; Provost and Vallee [5] motivated
the definition of this quantum metric by requiring that the distance induced by the inner
product in Hilbert state space be gauge independent with respect to the parameter space.
On the other hand, the imaginary part of the quantum geometric tensor gives rise to
the Berry curvature [6], which was given a formal geometric interpretation by Simon [7]
and is central in the study of adiabatic phenomena. The celebrated Berry phase appears
as the flux of the Berry curvature through the enclosed area of a circuit in parameter
space that is adiabatically traversed. The Berry phase was seen to possess a classical
counterpart, the Hannay angle [8], and the relation between them was established by
Berry through the semiclassical approximation of the wave function [9]. The Berry phase
was later generalized for a non-adiabatic evolution and for general quantum states in the
framework of the projective Hilbert space [10]. Likewise, the Fubini-Study distance was
shown to induce a measure of speed in the projective Hilbert space for a general evolution
[11, 12].
Nowadays, the quantum metric tensor and the Berry curvature have found a broad
range of applicability in quantum mechanics, as well as in other areas of physics like
condensed matter and information theory. For example, information theory concepts like
fidelity are deeply related to the quantum metric tensor and can be used to predict phase
transitions in quantum systems [13]. Also, divergences that appear in the components of
the quantum metric metric tensor are intimately related to quantum phase transitions
[14–20]; for instance, there is a complete study of the Riemannian geometry associated
with the spin XY chain in a transverse field [15, 21], where different types of singularities
in the metric and curvature are described, and experiments that can account for these
phenomena are suggested [21]. The quantum metric tensor has also great relevance in
the study of superfluids [22], Bogoliubov quasiparticles [23], and more recently, it has
proven to be an essential element in quantum computation [24–26] and complexity [27].
The geometric perspective has also been fruitful in the understanding of other branches
xv
xvi Introduction
of physics. For instance, in thermodynamics it has evolved since the proposals of Wein-
hold and Ruppeiner [28–31], until the more recent geometrothermodynamics [32, 33],
which has even been applied to black holes and cosmology. Therefore, the relation be-
tween geometric objects and physical properties has been successfully implemented and
has been found to possess a great predictive power. It also leads to a deeper insight of
the physical phenomena under consideration. In this spirit, different experiments have
been proposed to measure the quantum metric [34–38].
Another area of interest that departs from the previous topics is the study of N = 4
super Yang-Mills theory (SYM), which is central to theoretical high energy physics. Re-
cent developments in the integrability of N = 4 SYM have made possible the complete
calculation of all tree-level amplitudes as well as up to fourth order loops in perturbation
theory [39]. This was accomplished by using a duality symmetry in momentum space.
The integrability has its origins in the existence of generators of symmetries that trans-
form the fields in the internal space of the theory. In this context, in [40] it was raised
the question about the possibility of constructing a consistent relativistic quantum field
theory that preserves the analogous of the Runge-Lenz vector of the attractive Coulomb
potential in classical mechanics. The answer to this question is that it is indeed possible
and that the theory is precisely N = 4 SYM processed by a Higgs mechanism that gives
mass to some scalars fields.
In this work, our first purpose is to delve deeper into the study of the quantum geo-
metric tensor. To do this, we will relate and illustrate two approaches to its computation
and introduce a new formulation in terms of the Wigner function formalism. A second
purpose is the investigation of a classical analog of the quantum metric tensor. Our
motivation is that information from parameter space can be retrieved through classical
methods since quantum systems may possess features that are attainable through the
computation of classical geometric objects that have a similar structure as the quan-
tum ones. This will lead us to the application of the quantum and classical methods to
the analysis of many-body systems that are important in quantum optics and nuclear
physics. Our last purpose is to obtain a relativistic hydrogen-like theory from N = 4
SYM which will preserve the SO(4) symmetry that is lost in the usual minimal-coupling
prescription. This will motivate the consideration of a Higgs mechanism on the La-
grangian of N = 4 SYM, and then to the application of the enhanced symmetry to the
solution of the resulting equation.
The structure of the thesis is as follows. Chapter 1 provides an overview of the
fidelity approach to quantum phase transitions and summarizes some geometrical con-
cepts that are relevant for this work. Chapter 2 deals with the quantum construction of
the parameter space geometry. The Berry phase, connection, curvature, as well as the
quantum metric tensor and quantum geometric tensor are described. Two approaches
are introduced and are proved to be equivalent. In Chapter 3, we present a Wigner
function formulation of the Abelian and non-Abelian quantum geometric tensors and
complement the concepts with two examples. In Chapter 4, we introduce the Hannay
angle, connection and curvature, and propose the classical analog of the quantum metric
xvii
tensor. These classical objects are obtained from a semiclassical approximation of their
quantum counterparts and different expressions to compute them are presented. Chap-
ters 5 and 6 treat several examples using the two approaches to the quantum and classical
geometric quantities. In Chapters 7, 8 and 9, we study the quantum phase transitions in
the Dicke and Lipkin-Meshkov-Glick models through the use of the quantum metric ten-
sor, its classical analog, and their scalar curvatures. In Chapter 10, we introduce a Higgs
mechanism on the N = 4 super Yang-Mills theory that yields a relativistic equation for
the Coulomb potential that preserves the SO(4) symmetry, and thus, allows us to build
a relativistic Runge-Lenz vector to find the solutions. We illustrate the Kustaanheimo-
Stiefel duality between this system and the relativistic harmonic oscillator and relate
their constants of motion. Finally, some conclusions and directions for future research
are laid out in Chapter 11.

Chapter 1
Quantum phase transitions and
geometry
The goal of this chapter is to provide a brief account of the basic tools to understand the
fidelity approach to quantum phase transitions, as well as some notions of Riemannian
geometry that will be used throughout this work. These two apparently different subjects
will be seen to merge in the next chapter, where important concepts such as the Berry
curvature and the quantum metric tensor appear.
1.1 Fidelity approach to quantum phase transitions
A quantum phase transition is characterized by a change in the analytic properties
of the ground state. As opposed to classical phase transitions (thermodynamic phase
transitions), it is usually said that quantum phase transitions occur at zero temperature
to remark that they are not driven by thermal fluctuations, but by fluctuations associated
with the Heisenberg uncertainty principle [41]. In this sense, a quantum phase transition
can occur when the parameters that appear in the Hamiltonian are varied.
In the context of quantum many-body systems, the main focus is on second-order
phase transitions, also known as continuous phase transitions. In terms of the Ehrenfest
classification scheme, we say that an nth-order quantum phase transition occurs when
the nth derivative is the lowest order derivative which is discontinuous. Usually, second-
order phase transitions occur in many-body systems in the limit where the system’s
extension becomes infinite (thermodynamic limit), thus leading to a non-analyticity at
a critical value x = xc in the ground state energy. The non-analyticity could be either
the limiting case of an avoided level-crossing or an actual level-crossing [41].
Given that drastic changes in the ground state occur at a quantum phase transition,
we expect that a measure of the degree of similarity between quantum states can provide
a characterization of that phenomenon. In this respect, we can use a concept borrowed
from quantum information: the fidelity. This quantity measures the closeness between
two quantum states, yielding the value of unity if both states are the same, or zero in
1
2 Quantum phase transitions and geometry
case they are orthogonal. Given two pure quantum states, denoted as |ψ1〉 and |ψ2〉, the
fidelity F is the modulus of their overlap, i.e. [13],
F(ψ1, ψ2) := |〈ψ1|ψ2〉|. (1.1)
It is worth noting that unlike the overlap, the fidelity is the same regardless of the phases
of each state1. The fidelity has the following properties:
0 ≤ F(ψ1, ψ2) ≤ 1,
F(ψ1, ψ2) = F(ψ2, ψ1),
F(Ûψ1, Ûψ2) = F(ψ1, ψ2),
F(ψ1 ⊗ ϕ1, ψ2 ⊗ ϕ2) = F(ψ1, ψ2)F(ϕ1, ϕ2), (1.2)
where Û represents a unitary transformation and |ψ〉, |ϕ〉 are the states of different
subsystems. Fidelity can also be used to compare mixed states described by the density
matrices ρ̂1 and ρ̂2. The most widely-used definition in this case is the Uhlmann fidelity
[44] given by
F(ρ1, ρ2) := Tr
√
√
ρ̂1ρ̂2
√
ρ̂1. (1.3)
This definition satisfies the properties (1.2). For pure states, we have ρ̂1 = |ψ1〉〈ψ1| and
ρ̂2 = |ψ2〉〈ψ2|, and the Uhlmann fidelity reduces to (1.1).
A closely related concept to fidelity is the Fubini-Study distance [45, 46]. Consider the
quantum states |ψ1〉 and |ψ2〉, then the Fubini-Study distance between them is defined
as
dFS(ψ1, ψ2) :=
√
2− 2|〈ψ1|ψ2〉| =
√
2− 2F(ψ1, ψ2). (1.4)
It can be shown that it satisfies the three axioms of a distance function [1]:
dFS(ψ1, ψ2) = 0 ⇐⇒ |ψ1〉 = |ψ2〉,
dFS(ψ2, ψ1) = dFS(ψ1, ψ2),
dFS(ψ1, ψ3) ≤ dFS(ψ1, ψ2) + dFS(ψ2, ψ3). (1.5)
To gain more intuition about the fidelity and the Fubini-Study distance, we can easily
establish an analogy with basic concepts of Euclidean geometry. Let x and y be two
unit vectors in R
2. The Euclidean distance between these two vectors can be obtained
as
||x− y|| =
√
(x− y) · (x− y) =
√
2− 2x · y, (1.6)
where x·y denotes the scalar product. If θ is the angle between the vectors, we know that
x ·y = cos θ. Therefore, comparing (1.6) with (1.4), we see that the fidelity is analogous
to the cosine of the angle between the vectors, whereas the Fubini-Study distance is
analogous to the Euclidean distance.
1There is an alternative definition of fidelity given by |〈ψ1|ψ2〉|2 (see [42, 43]). However, in this work
we will employ (1.1) only
Elements of Riemannian geometry 3
Now, when the states under consideration in the evaluation of the Fubini-Study
distance (1.4) are infinitesimally separated such that |ψ1〉 = |ψ〉 and |ψ2〉 = |ψ〉 + |δψ〉,
then the Fubini-Study metric, also known as the quantum metric tensor, is obtained. We
will see the emergence of this object in the next chapter following the work of Provost
and Valle [5]. It is worth mentioning that the Fubini-Study metric can also be shown to
appear when the Fisher metric, which is used in information theory to provide a measure
of distance in a statistical manifold, is extended to a complex projective space [2].
Finally, regarding the generalization to mixed states, if equation (1.4) is evaluated
with the Ulhmann fidelity (1.3) instead of (1.1), then we obtain the so-called Bures
distance [47]. The infinitesimal expansion of this distance results in the Bures metric,
which is in general not easy to calculate due to the presence of the symmetric logarithm
derivative operator [3].
1.2 Elements of Riemannian geometry
We now provide a brief account of the basic geometric tools that will later be useful for
analyzing the parameter space of a given system. The differential-geometric construction
begins with the concept of manifold. A (differentiable) manifold M of dimension N is
a topological space that locally resembles R
N . In this sense, we can assign coordinates
xα = {xiα}, (i = 1, ..., N) to an open subset Aα ⊂ M that provide a one-to-one map
φα : Aα → R
N such that the image φα(Aα) is open in R
N . The pair {Aα, φα} is called a
chart. We further require that the union of all the charts
⋃
αAα cover the manifold, and
that where two charts intersect, the map (φα ◦ φ−1β ) is infinitely differentiable, meaning
that the charts are smoothly sewn together [48].
At this point, we introduce the Einstein summation convention to write compact
expressions for sums by omitting the symbol Σ. In this way, when the same index
appears two times, one as a subscript and the other as a superscript, a sum is implied.
We call this a contraction. For example, the expression V iωi means
∑N
i=1 V
iωi.
We now proceed to the introduction of tensors on a manifold. We begin with the con-
cept of a vector. A (contravariant) vector at a point P is an object whose N components
{V i} transform as
V i → V ′i =
∂x′i
∂xj
V j (1.7)
under a change of coordinates x→ x′ = x′(x). Here, the derivatives are evaluated at P .
The set of all the vectors that are defined at P is known as the tangent space TP . The
(coordinate) basis of this vector space is the set of directional derivatives at P , therefore,
a vector V is written as V = V i∂i, where ∂i := ∂
∂xi
.
Analogously, we define a covariant vector, or one-form, as an object whose N com-
ponents {ωi} transform as
ωi → ω′i =
∂xj
∂x′i
ωj (1.8)
4 Quantum phase transitions and geometry
under the change of coordinates x→ x′ = x′(x). The set of all the one-forms at a given
point P is called the cotangent space T ∗p and its (coordinate) basis is formed by the
differential displacements dxi, thus, a one-form is expanded as ω = ωidx
i.
More generally, we define a tensor of rank (m,n) as an object whose components
T i1···imj1···jn transform as
T i1···imj1···jn → T ′i1···imj1···jn =
∂x′i1
∂xk1
· · · ∂x
′im
∂xkm
∂xl1
∂x′j1
· · · ∂x
ln
∂x′jn
T k1···kml1···ln (1.9)
under the change of coordinates x→ x′ = x′(x). In this case, the tensor is expressed as
T = T i1···imj1···jn∂i1 ⊗ · · · ⊗ ∂im ⊗ dxj1 ⊗ · · · ⊗ dxjn . We readily recognize that vectors are
(1, 0) tensors and one-forms are (0, 1) tensors. It is important to note that an object with
no free indices, like the contraction V iωi, is invariant under coordinate transformations
due to the properties (1.7) and (1.8). Such an object is called a scalar and is a (0, 0)
tensor.
A fundamental object in Riemannian geometry is the metric tensor gij(x), which
allows us to measure distances on the manifold M. The metric is a symmetric (0, 2)
tensor, hence, its components transform as
gij → g′ij =
∂xk
∂x′i
∂xl
∂x′j
gkl (1.10)
under the change x → x′ = x′(x). The metric tensor defines an inner product over M
which is invariant under coordinate transformations: V ·W = gijV
iW j. This, in turn,
allows us to define the (squared) norm ||V ||2 = V · V , and as a consequence, to measure
distances. The metric is also non-degenerate, i.e., V ·W = 0 for all V only if W = 0.
In particular, this means that the matrix with elements gij is invertible at every point.
Denoting the components of the inverse metric as gij, we have that gikgkj = δij. With
this at hand, we can find the infinitesimal (squared) distance between two points whose
coordinates are x and x+ dx as
ds2 = gij(x)dx
idxj. (1.11)
Other tensors of interest are differential forms, or n-forms, which are completely
antisymmetric (0, n) tensors. To discuss their properties, we first introduce the wedge
product defined as
dxi ∧ dxj =
1
2
(dxi ⊗ dxj − dxj ⊗ dxi) = −dxj ∧ dxi. (1.12)
Naturally, dxi ∧ dxi = 0. Under the coordinate transformation x → x′ = x′(x), we see
that dxi ∧ dxj transforms as [49]
dxi ∧ dxj → dx′i ∧ dx′j =
(
∂x′i
∂xk
∂x′j
∂xl
− ∂x′i
∂xl
∂x′j
∂xk
)
dxk ∧ dxl, (1.13)
Elements of Riemannian geometry 5
which is precisely the transformation rule of an area element. An n-form ω defined on
the manifold M is then expressed as
ω =
1
n!
ωi1···indx
i1 ∧ · · · ∧ dxin , (1.14)
where the coefficients ωi1···in are completely antisymmetric. The dimension of the vector
space Λn(x), which contains all the n-forms at a point P with coordinates x, is
(
N
n
)
=
N !
n!(N−n)! . The wedge product can be used to operate between differential forms. If ω is
an m-form and η is an n-form, then
ω ∧ η =
(m+ n)!
m!n!
ω[i1···imηim+1···im+n]dx
i1 ∧ · · · ∧ dxim ∧ dxim+1 ∧ · · · ∧ dxim+n (1.15)
gives as a result an (m + n)-form, where [· · ·] performs the antisymmetrization of the
indices contained inside the brackets. Also,
ω ∧ η = (−1)mnη ∧ ω. (1.16)
The n-forms can be differentiated with the aid of the exterior derivative. This operation
takes n-forms into (n+ 1)-forms as
d
(
1
n!
ωi1···indx
i1 ∧ · · · ∧ dxin
)
= (n+ 1)∂[in+1ωi1···in]dx
in+1 ∧ dxi1 ∧ · · · ∧ dxin , (1.17)
The exterior derivative of the wedge product of an m-form ω and an n-form η is
d(ω ∧ η) = dω ∧ η + (−1)mω ∧ dη. (1.18)
Furthermore, d(dω) = 0. The exterior derivative allows us to state an important result
that generalizes the fundamental theorem of calculus: the Stokes theorem. It says that
given an (n− 1)-form ω and an orientable n-dimensional manifold Ω, then
∫
Ω
dω =
∫
∂Ω
ω, (1.19)
where ∂Ω is the boundary of Ω.
Finally, we can characterize the local behavior of our manifold M with the Riemann
tensor Ri
jkl. This object can be constructed from second derivatives of the metric
combined in a specific way and it contains all the information about the curvature in
the vicinity of a given point. Essentially, it arises from the commutator of the covariant
derivatives which are formed from a metric-compatible and symmetric connection, i.e.,
the Levi-Civita connection (for more details, see [48]). In two dimensions, which will be
the case for the models that we analyze in Chapters 7, 8, and 9, the Riemann tensor has
only one independent component, and thus, its contraction R = gijRk
ikj known as the
6 Quantum phase transitions and geometry
scalar curvature (or Ricci scalar) provides an invariant measure of the curvature. Using
local coordinates x = (x1, x2), the scalar curvature can be computed as [50]
R =
1√
g
(A+ B) (1.20)
where
A =
∂
∂x1
[
1√
g
(
g12
g11
∂g11
∂x2
− ∂g22
∂x1
)]
,
B =
∂
∂x2
[
1√
g
(
2
∂g12
∂x1
− ∂g11
∂x2
− g12
g11
∂g11
∂x1
)]
, (1.21)
and g is the determinant of the metric.
As a first example to illustrate the concept of scalar curvature, let our manifold M
be a sphere with radius a. We take as our coordinates the polar and azimuthal angles,
hence, x = {xi} = (θ, φ), (i = 1, 2). The infinitesimal squared distance between two
points on the sphere is given by
ds2 = a2dθ2 + a2 sin2 θ dφ2, (1.22)
from which we can extract the metric tensor
gij =
(
a2 0
0 a2 sin2 θ
)
. (1.23)
Using equation (1.20), we find the scalar curvature
R =
2
a2
. (1.24)
This result indicates that a sphere has a positive constant curvature. Furthermore,
increasing the radius makes the curvature smaller, as we intuitively expect, and in the
limit a→ ∞ the scalar curvature approaches to zero, which is the curvature of a plane.
As our second example, we consider the Lobachevsky space H2 [51]. Taking as our
coordinates x = {xi} = (ρ, θ), (i = 1, 2), the distance element is
ds2 = a2(dρ2 + sinh2 ρ dθ2), (1.25)
where a > 0 is a constant. The metric tensor is
gij =
(
a2 0
0 a2 sinh2 ρ
)
, (1.26)
and its resulting scalar curvature is
R = − 2
a2
. (1.27)
The negative scalar curvature means that this manifold has hyperbolic geometry, thus
having a saddle-like shape at each point.
Chapter 2
Quantum geometric tensor
In this chapter, we summarize the basic results concerning the emerge of the Berry
phase [6] and the quantum metric tensor, and the associated geometric structure that
naturally appears. We remark the important feature that all the geometric quantities
can be rewritten in terms of the generators of translations in parameter space, and
note that this will later prove to possess great relevance. Then, we introduce the path
integral approach to the quantum geometric tensor and prove our first result: that the
expressions obtained under this formulation are the same that those obtained with the
generator approach. Finally, we provide the different ways of computing the objects that
encode the geometry of the parameter space.
2.1 The Berry phase, connection and curvature
The notion of adiabaticity is usually introduced through a system with a Hamiltonian
Ĥ(x) that depends on some parameters x(t) = {xi}, i = 1, ...,M , which at the same
time have a dependence on time1. We say that we are in the adiabatic regime when
the time it takes the parameters to change appreciably is much longer compared to the
natural time scale of the system [4]. Moreover, the parameters change in a form which
is independent of the dynamics of the system. The adiabatic theorem states that if a
quantum system begins in the Hamiltonian’s eigenstate |n(x)〉, then after an adiabatic
evolution where the parameters change from x to x+ dx, the system will remain in the
same state eigenstate, assuming that the nth energy level is non degenerate. In this
work, we restrict ourselves to the consideration of the adiabatic regime. Generalizations
of some of the geometric structures that will be introduced and some aspects of the
parameters’ choice can be found in [10] and [52], respectively.
The time evolution in the adiabatic approximation is obtained by solving the instan-
taneous Schrödinger equation
Ĥ(x)|n(x)〉 = En(x)|n(x)〉. (2.1)
1Along this work, the parameters x(t) will be denoted as x to simplify notation.
7
8 Quantum geometric tensor
The result is that during evolution, the state will acquire a dynamic phase
αn(t) = exp

− i
~
t
∫
0
dt′En(t
′)

 , (2.2)
and a so-called geometric phase
γn(t) =
t
∫
0
dt′ i
〈
n
∣
∣
∣
∣
dn
dt′
〉
=
t
∫
0
dt′ i〈n|ṅ〉. (2.3)
If the system is taken through a closed curve C : {xi = xi(t) | t ∈ [0, T ]} in parameter
space, the geometric phase can be written as
γn(C) =
∮
C
dxiA
(n)
i , (2.4)
where
A
(n)
i = i
〈
n
∣
∣
∣
∣
∂n
∂xi
〉
= i〈n|∂in〉. (2.5)
We see that this introduces a one-form A(n) that in local coordinates is expressed as
A(n) = A
(n)
i dxi (we use the Einstein summation convention), and thus,
γn(C) =
∮
C
A(n). (2.6)
This geometric phase γn(C) is known as Berry phase [6]. Using Stokes theorem, we can
cast it in the form
γn(C) =
∫
Σ
F (n), (2.7)
with F (n) = dA(n) and Σ a surface such that C = ∂Σ. In local coordinates, the two-form
F (n) is
F (n) =
1
2
F
(n)
ij dxi ∧ dxj, (2.8)
where
F
(n)
ij = ∂iA
(n)
j − ∂jA
(n)
i . (2.9)
We assume that we are working with an orthonormal set of states such that 〈m|n〉 = δmn.
Taking the derivative of this condition with respect to xi, we see that
〈∂im|n〉 = −〈m|∂in〉, (2.10)
and letting m = n, we conclude that
Re〈n|∂in〉 = 0. (2.11)
Quantum metric tensor 9
This indicates that A(n)
i = i〈n|∂in〉 is real, hence,
A
(n)
i = −Im〈n|∂in〉. (2.12)
Of course, this also entails that F (n)
ij is real and can be written as
F
(n)
ij = i (〈∂in|∂jn〉 − 〈∂jn|∂in〉) = −Im (〈∂in|∂jn〉 − 〈∂jn|∂in〉) . (2.13)
We know that two states differing by a phase are equivalent, so the state |n′〉 = eiλn |n〉
yields the same expectation value as |n〉 for any observable. We see that both A
(n)
i ,
and F
(n)
ij , and thus the Berry phase, remain invariant under this phase transformation.
However, if we allow the phase to depend on the parameters x, i.e., we perform the local
phase transformation
|n〉 → |n′〉 = eiλn(x)|n〉, (2.14)
we see that
A
(n)
i → A
′(n)
i = A
(n)
i − ∂iλn, (2.15)
F
(n)
ij → F
′(n)
ij = F
(n)
ij . (2.16)
This transformation is reminiscent of the Abelian U(1) gauge transformation of electro-
magnetism, thus A(n) is really a connection one-form (Berry connection), and F (n) is
its curvature two-form (Berry curvature) [7, 53]. Notice that the Berry phase remains
invariant too.
2.2 Quantum metric tensor
We can easily see that the distance in Hilbert space naturally induces a metric. The
square of this distance between quantum spaces infinitesimally close is given by
dL2(n(x), n(x+ δx)) = 〈n(x+ δx)− n(x)|n(x+ δx)− n(x)〉. (2.17)
Expanding this function to second order, we see that
dL2(n(x), n(x+ δx)) = γ
(n)
ij δx
iδxj, (2.18)
where γ(n)ij = 1
2
(〈∂in|∂jn〉+ 〈∂jn|∂in〉). This might lead us into thinking that the sym-
metric matrix γ(n)ij is the natural choice of metric for our parameter space. However, this
“metric” is not invariant under the U(1) gauge transformation
|n〉 → |n′〉 = eiλn(x)|n〉. (2.19)
Provost and Valle [5] noticed that another term can be added to γ(n)ij such that now we
have a gauge invariant metric, also known as the quantum metric tensor:
g
(n)
ij = Re (〈∂in|∂jn〉 − 〈∂in|n〉〈n|∂jn〉) , (2.20)
10 Quantum geometric tensor
and hence, the appropriate (squared) distance ds2 between the two infinitesimally near
quantum states is now given by
ds2 = g
(n)
ij δx
iδxj, (2.21)
which can also be written in terms of the quantum fidelity F(x, x + δx) = |〈n(x +
δx)|n(x)〉| as
ds2 = 2− 2F(x, x+ δx). (2.22)
Therefore, as anticipated in Chapter 1, ds =
√
2− 2F(x, x+ δx) is the infinitesimal
version of the Fubini-Study distance which is why g
(n)
ij is also called the Fubini-Study
metric.
The fact that g(n)ij is actually a tensor of rank (0, 2) is easily proved, since it is evident
from (2.20) that under the change of parameters xi → x′i = x′i(x), it transforms as
g
′(n)
ij =
∂xk
∂x′i
∂xl
∂x′j
g
(n)
kl . (2.23)
Besides, it is positive semidefinite [4], so that the length of any curve C on the parameter
space is
I(C) =
∫
C
√
g
(n)
ij δx
iδxj ≥ 0. (2.24)
2.3 Abelian and non-Abelian quantum geometric ten-
sor
The (Abelian) quantum geometric tensor for the nth non-degenerate quantum state is
defined as
G
(n)
ij = 〈∂in|∂jn〉 − 〈∂in|n〉〈n|∂jn〉. (2.25)
It is easily seen that it can be separated as
G
(n)
ij = g
(n)
ij − i
2
F
(n)
ij , (2.26)
so that the metric and the Berry curvature can be extracted from the quantum geometric
tensor as
g
(n)
ij = ReG
(n)
ij , F
(n)
ij = −2ImG
(n)
ij . (2.27)
We can as well introduce the non-Abelian quantum geometric tensor [54]. This
object is a generalization of the quantum geometric tensor (2.25) to degenerate states of
a quantum system. Let us consider a Hamiltonian Ĥ that depends on a set ofM adiabatic
parameters x = {xi}, and suppose that the nth energy level is gn times degenerate, i.e.,
Ĥ(x)|nI(x)〉 = En(x)|nI(x)〉, I = 1, 2, ..., gn, (2.28)
Abelian and non-Abelian quantum geometric tensor 11
Assuming that we have chosen an orthonormal set of eigenstates on the nth subspace,
〈nI(x)|nJ(x)〉 = δIJ , the non-Abelian quantum geometric tensor is defined as
G
(n)
ijIJ = 〈∂inI |
(
1−
gn
∑
K=1
|nK〉〈nK |
)
|∂jnJ〉. (2.29)
Of course, in the non-degenerate case gn = 1, this non-Abelian tensor reduces to the
Abelian one (2.25). Under the unitary phase transformation
|nI(x)〉 → |n′I(x)〉 =
gn
∑
J=1
|nJ(x)〉U (n)
JI (x), (2.30)
where U (n)
JI (x) are the entries of a unitary matrix of dimension gn × gn, the geometric
tensor (2.29) transforms as
G
(n)
ijIJ → G
(n)′
ijIJ =
gn
∑
K,L=1
U
(n)∗
KI G
(n)
ijKLU
(n)
LJ . (2.31)
From this tensor, we can extract the non-Abelian quantum metric tensor
g
(n)
ijIJ =
1
2
(
G
(n)
ijIJ +G
(n)∗
ijJI
)
, (2.32)
and the non-Abelian Wilczek-Zee curvature [55]
F
(n)
ijIJ = i
(
G
(n)
ijIJ −G
(n)∗
ijJI
)
. (2.33)
In the degenerate case, the application of the adiabatic theorem gives as a result that
at the end of a cyclic evolution, the system will remain in the nth subspace acquiring a
dynamical phase (the same as in the Abelian case), and a geometric phase (Wilczek-Zee
factor [55]) which now is
V = P exp

i
∮
C
A(n)

 , (2.34)
where P denotes path-ordering and A(n) is the matrix one-form
A(n) = A
(n)
i dxi (2.35)
with elements
A
(n)
iIJ = i〈nI |∂inJ〉. (2.36)
It is easily found that under the unitary transformation (2.30), the one-form A(n) trans-
forms as
A(n) → A′(n) = U (n)†A(n)U (n) + iU (n)†(dU (n)), (2.37)
12 Quantum geometric tensor
or in component form,
A
(n)
i → A
′(n)
i = U (n)†A
(n)
i U (n) + iU (n)†(∂iU
(n)). (2.38)
This is precisely the transformation associated with a connection of the non-Abelian
gauge group SU(N). The corresponding non-Abelian curvature two-form F (n) is written
as
F (n) =
1
2
F
(n)
ij dxi ∧ dxj, (2.39)
where the matrix elements of F (n)
ij are given by (2.33). In differential-form notation, the
Wilczek-Zee curvature takes on the familiar form
F (n) = dA(n) − iA(n) ∧ A(n), (2.40)
or, in terms of components,
F
(n)
ij = ∂iA
(n)
j − ∂jA
(n)
i − i[A
(n)
i , A
(n)
j ]. (2.41)
The Wilczek-Zee curvature is not gauge-invariant, as opposed to the Abelian case; it
transforms as
F (n) → F ′(n) = U (n)†F (n)U (n). (2.42)
However, due to the cyclic property of the trace, TrF (n) is gauge-invariant.
From now on, when using the term quantum geometric tensor, we only refer to the
Abelian tensor (2.25), except in Chapter 3, where both the Abelian and non-Abelian
geometric tensors are treated.
2.4 Generator approach to the quantum geometric ten-
sor
Now, we introduce the concept of generator of translations in parameter space, also
known as adiabatic gauge potentials [56]. First, note that the displaced state |n(x+δx)〉
is given, to first order in δx by
|n(x+ δx)〉 = |n(x)〉+ ∂i|n(x)〉δxi, (2.43)
so we can rewrite this displacement in terms of the unitary translation operator T̂ (δx) =
1 − i
~
Ĝiδx
i. Notice that the unitarity of T̂ (δx) implies that all Ĝi, which we shall call
generators of translations in parameter space (or simply, generators), are Hermitian.
Thus, our Ĝi will satisfy the differential equation
i~∂i|n(x)〉 = Ĝi|n(x)〉. (2.44)
Path integral approach to the quantum geometric tensor 13
This simple, but crucial observation, enables us to rewrite all the geometric quantities
we have defined so far in terms of the Ĝi:
A
(n)
i =
1
~
〈Ĝi〉n, (2.45)
F
(n)
ij =
i
~2
〈[Ĝi, Ĝj]〉n = Im
(
− 1
~2
〈[Ĝi, Ĝj]〉n
)
, (2.46)
g
(n)
ij =
1
~2
(
1
2
〈{Ĝi, Ĝj}〉n − 〈Ĝi〉n〈Ĝj〉n
)
= Re
[
1
~2
(
〈ĜiĜj〉n − 〈Ĝi〉n〈Ĝj〉n
)
]
, (2.47)
where 〈X̂〉n = 〈n|X̂|n〉 is the expectation value of the operator X̂ with respect to the
state |n〉. Notice that the Hermiticity of the generators is consistent with the reality of
the Berry connection, the curvature, and the metric. Furthermore, if we define
∆Ĝi = Ĝi − 〈Ĝi〉n, (2.48)
and ∆Ĝ = ∆Ĝiδx
i, we can see that the distance takes the form
dL2 = Re
[
1
~2
〈(∆Ĝ)2〉n
]
, (2.49)
i.e., the distance is the variance of the complete generator Ĝ = Ĝiδx
i. Finally, we note
that by virtue of (2.44), the action of the operator ∆Ĝj on the state |n〉 can be written
as
∆Ĝj|n〉 =
(
i~∂j − 〈Ĝj〉n
)
|n〉 = i~
(
∂j + iA
(n)
j
)
|n〉, (2.50)
which resembles the structure of a covariant derivative D(n)
j = ∂j+ iA
(n)
j with connection
A
(n)
j .
2.5 Path integral approach to the quantum geometric
tensor
In this section, we briefly present a path integral approach to the quantum geometric
tensor. This approach was first introduced in [57] for the quantum metric tensor in the
context of gauge-gravity duality, and was later generalized to include both the metric
and the Berry curvature in [58].
Consider a quantum system in the time interval t ∈ (−∞, 0) which is described
through the path integral formulation by a Hamiltonian H(q(t), p(t); x), where q = {qa}
and p = {pa}, a = 1, ..., N are the coordinates and the momenta, and x = {xi} with
i = 1, ...,M is a set of M adiabatic parameters. Suppose that at t = 0, a perturbation is
turned on such that the system will now be described by a Hamiltonian H ′ = H+Oiδx
i
14 Quantum geometric tensor
during the time interval t ∈ (0,∞). The functions Oi(t) are known as Hamiltonian
deformations and are given by
Oi(t) =
(
∂H
∂xi
)
q,p
. (2.51)
In order to compare the ground states |0〉 and |0′〉 of H and H ′, respectively, we use
the fidelity F(x, x+ δx) = |〈0′|0〉|. A treatment of the kernel 〈q′,∞|q,−∞〉 in the path
integral formalism leads to [58]
〈0′|0〉 =
〈
exp
(
i
~
∫∞
0
dtOiδx
i
) 〉
0
〈
exp
(
i
~
∫∞
−∞ dtOiδxi
)
〉1/2
0
, (2.52)
where the expectation value 〈Â〉0 is taken in the ground state of H as
〈Â〉0 =
1
Z0
∫
DqDpA(q, p; x) exp


i
~
∞
∫
−∞
dt (pq̇ −H)

 , (2.53)
and Z0 =
∫
DqDp exp
[
i
~
∫∞
−∞ dt (pq̇ −H)
]
. An expansion of the fidelity F(x, x+ δx) to
second order yields [58]
F(x, x+ δx) = 1− 1
2
G
(0)
ij (x)δx
iδxj, (2.54)
where
G
(0)
ij (x) = − 1
~2
0
∫
−∞
dt1
∞
∫
0
dt2
(
〈Ôi(t1)Ôj(t2)〉0 − 〈Ôi(t1)〉0〈Ôj(t2)〉0
)
(2.55)
is the quantum geometric tensor for the ground state |0〉. Assuming time-reversal sym-
metry for the two-point functions, 〈Ôi(−t1)Ôj(−t2)〉0 = 〈Ôi(t1)Ôj(t2)〉0, the quantum
metric tensor can be obtained as the real part of (2.55):
g
(0)
ij (x) = − 1
~2
0
∫
−∞
dt1
∞
∫
0
dt2
(
1
2
〈{Ôi(t1), Ôj(t2)}〉0 − 〈Ôi(t1)〉0〈Ôj(t2)〉0
)
, (2.56)
where {·, ·} is the anticommutator. On the other hand, taking the imaginary part of
(2.55) as F (0)
ij (x) = −2ImG
(0)
ij (x), we get the Berry curvature:
F
(0)
ij (x) =
1
i~2
0
∫
−∞
dt1
∞
∫
0
dt2 〈[Ôi(t1), Ôj(t2)]〉0, (2.57)
Equivalence between the two approaches 15
with [·, ·] the commutator.
We must mention that equation (2.55) was obtained under a path integral formula-
tion, however, it may be as well evaluated traditionally using operators acting on states
of the Hilbert space of the system. As a matter of fact, it turns out that (2.55) can be
used to find the quantum geometric tensor not only for the ground state, but for any
excited state |n〉 as follows:
G
(n)
ij (x) = − 1
~2
0
∫
−∞
dt1
∞
∫
0
dt2
(
〈Ôi(t1)Ôj(t2)〉n − 〈Ôi(t1)〉n〈Ôj(t2)〉n
)
. (2.58)
The expectation values that appear here are naturally taken in the nth eigenstate of the
Hamiltonian, i.e., 〈·〉n = 〈n| · |n〉. The following section is devoted to prove this fact and
to establish the relation between expressions (2.25) and (2.58).
2.6 Equivalence between the two approaches
Here, we present the first result of our work: the proof that equations (2.25) and (2.58)
are indeed the same. This was not proved in reference [58]; the equivalence between
these two approaches was only shown through some examples.
Let us start with equation (2.58) and try to get to (2.25). First, we move to the
Schrödinger picture where the operators are time-independent. This means that the
Heisenberg operator Ôi(t) will be written as Ôi(t) = e
i
~
ĤtÔie
− i
~
Ĥt, where Ôi is the
Schrödinger operator Ôi(t = 0). From the second term in the above expression, we
clearly see that 〈Ôi(t1)〉n = 〈n|e i
~
Ĥt1Ôie
− i
~
Ĥt1 |n〉 = 〈n|Ôi|n〉, since we can apply the first
exponential to the left and the second to the right, canceling each other out. Now, we
use the same manipulation and we find that the first term takes the form
〈Ôi(t1)Ôj(t2)〉n = e−
i
~
En(t2−t1)〈n|Ôie
− i
~
Ĥt1e
i
~
Ĥt2Ôj|n〉. (2.59)
We now insert the identity operator 1̂ =
∑
m |m〉〈m| between the two exponentials inside
the bracket, which results in
〈Ôi(t1)Ôj(t2)〉n =
∑
m
e
i
~
(Em−En)(t2−t1)〈n|Ôi|m〉〈m|Ôj|n〉. (2.60)
The sum in this expression runs over all the allowed values of m, however, if we split the
sum as
∑
m 6=n e
i
~
(Em−En)(t2−t1)〈n|Ôi|m〉〈m|Ôj|n〉 + 〈n|Ôi|n〉〈n|Ôj|n〉, it is obvious that
the integrand in (2.58) reduces to
〈Ôi(t1)Ôj(t2)〉n − 〈Ôi(t1)〉n〈Ôj(t2)〉n =
∑
m 6=n
e
i
~
(Em−En)(t2−t1)〈n|Ôi|m〉〈m|Ôj|n〉, (2.61)
16 Quantum geometric tensor
and thus, the quantum geometric tensor takes the form
G
(n)
ij = − 1
~2
∑
m 6=n


0
∫
−∞
dt1
∞
∫
0
dt2 e
i
~
(Em−En)(t2−t1)

 〈n|Ôi|m〉〈m|Ôj|n〉. (2.62)
We have completely isolated the time-dependence and now we wish to integrate it. We
notice that the integrand is oscillatory, and therefore, it is not well-defined in the limits
−∞ and ∞, however, it is possible to apply a convenient regularization that is often
used in field theory. Having in mind the ranges of t1 and t2, we establish the following
prescription to make sense of the integral:
0
∫
−∞
dt1
∞
∫
0
dt2 e
i
~
(Em−En)(t2−t1) = lim
ǫ→0+
0
∫
−∞
dt1
∞
∫
0
dt2 e
i
~
(Em−En+iǫ)(t2−t1) = − ~
2
(Em − En)2
.
(2.63)
This prescription will be used along this work multiple times. Taking this result into
account, (2.62) turns into
G
(n)
ij =
∑
m 6=n
〈n|Ôi|m〉〈m|Ôj|n〉
(Em − En)2
. (2.64)
This is known as the perturbative form of the quantum geometric tensor [13]. This
expression is relevant because divergences are explicitly seen to occur when level-crossing
takes place, i.e., when there is some value of the parameters x = x∗ such that Em(x∗) =
En(x
∗), which may be a signature of a quantum phase transition [41]. In (2.64), we have
removed the time-dependence of the operators, but a disadvantage of this formulation is
that we now have to take the expectation values of the Hamiltonian deformations over
all the quantum states of the system, which is not in general an easy task.
From (2.64), it is straightforward to arrive at the quantum geometric tensor that
Provost and Vallee [5] first obtained. Differentiating the Schrödinger equation, we get
that 〈∂in|m〉 = 〈n|∂iĤ|m〉
En−Em
and 〈m|∂jn〉 = 〈m|∂jĤ|n〉
En−Em
for m 6= n, which leads to
G
(n)
ij =
∑
m 6=n
〈∂in|m〉〈m|∂jn〉 =
∑
m
〈∂in|m〉〈m|∂jn〉 − 〈∂in|n〉〈n|∂jn〉. (2.65)
We have added and subtracted one term to complete the sum. In this expression, we
readily recognize the identity operator 1̂ =
∑
m |m〉〈m|, hence, the quantum geometric
tensor finally takes the form
G
(n)
ij = 〈∂in|∂jn〉 − 〈∂in|n〉〈n|∂jn〉, (2.66)
which is just equation (2.25).
Summarizing, the expressions to compute the quantum geometric tensor for the nth
state are:
Equivalence between the two approaches 17
1. Provost and Vallee
G
(n)
ij (x) = 〈∂in|∂jn〉 − 〈∂in|n〉〈n|∂jn〉. (2.67)
2. Generator approach
G
(n)
ij (x) =
1
~2
(
〈ĜiĜj〉n − 〈Ĝi〉n〈Ĝj〉n
)
. (2.68)
3. Perturbative form
G
(n)
ij (x) =
∑
m 6=n
〈n|Ôi|m〉〈m|Ôj|n〉
(Em − En)2
. (2.69)
4. Path integral approach
G
(n)
ij (x) = − 1
~2
0
∫
−∞
dt1
∞
∫
0
dt2
(
〈Ôi(t1)Ôj(t2)〉n − 〈Ôi(t1)〉n〈Ôj(t2)〉n
)
. (2.70)
Each of these expressions possesses advantages and disadvantages, and it depends on
the system at hand to decide which one will be used. Naturally, once the quantum
geometric tensor has been obtained, the quantum metric and the Berry curvature can be
extracted from its real and imaginary parts as (2.27). In the next chapter, we introduce
yet another approach to arrive at the quantum geometric tensor in both the Abelian and
non-Abelian cases.

Chapter 3
Phase space formulation of the
quantum geometric tensor
In this chapter, we present another important result of this work: the Wigner function
formulation of the Berry and Wilczek-Zee connections, as well as of the Abelian and non-
Abelian quantum geometric tensor. We briefly describe the Wigner function formalism,
and then, tackle the issue of incorporating in this description the geometric objects
that describe the parameter space. As an illustration of the phase space formulation, we
provide two illustrative examples: the generalized harmonic oscillator and two symmetric
coupled harmonic oscillators. The results presented here have been published in [59].
3.1 Wigner function formalism
Let us first introduce the basic elements of the phase space formulation of quantum
mechanics, also known as the Wigner function formalism [60–63]. Given an operator
Q̂(q̂, p̂; x) that depends on q̂ = {q̂a}, p̂ = {p̂a}, (a = 1, ..., N) and a set of parameters
x = {xi}, (i = 1, ...,M), its Wigner transform is
Q(q, p; x) =
∞
∫
−∞
dNy e−
ip·y
~
〈
q +
y
2
∣
∣
∣
∣
Q̂(q̂, p̂; x)
∣
∣
∣
∣
q − y
2
〉
, (3.1)
where the variables q = {qa}, p = {pa} are the eigenvalues of the operators q̂ and p̂,
respectively, and p · y =
∑N
a=1 pay
a. The Wigner transform has the following property
of composition of two operators Q̂1 and Q̂2:
Tr(Q̂1Q̂2) =
1
(2π~)N
∞
∫
−∞
dNqdNpQ1Q2, (3.2)
where Q1 and Q2 are the corresponding Wigner transforms of Q̂1 and Q̂2.
19
20 Phase space formulation of the quantum geometric tensor
The Wigner function Wn(q, p; x) of a quantum state |n(x)〉 is defined as the Wigner
transform of the density operator ρ̂n(x) = |n(x)〉〈n(x)| modulo a constant, i.e.,
Wn(q, p; x) =
1
(2π~)N
∞
∫
−∞
dNy e−
ip·y
~
〈
q +
y
2
∣
∣
∣
∣
n(x)
〉〈
n(x)
∣
∣
∣
∣
q − y
2
〉
=
1
(2π~)N
∞
∫
−∞
dNy e−
ip·y
~ ψn
(
q +
y
2
; x
)
ψ∗n
(
q − y
2
; x
)
. (3.3)
Using (3.2) with Q̂1 → ρ̂n and Q̂2 → Ô, we immediately see that the expectation value
of the operator Ô can be written as
〈Ô〉n = Tr(ρ̂nÔ) =
∞
∫
−∞
dNqdNpWn(q, p; x)O(q, p; x), (3.4)
which resembles the average of the function O(q, p; x) over the phase space variables.
This is why the Wigner function is dubbed as a quasiprobability distribution, since it is
real but not positive everywhere. Two last properties that we need are
∞
∫
−∞
dNqdNpWn(q, p; x) = 1, (3.5)
and ∞
∫
−∞
dNqdNpW 2
n(q, p; x) =
1
(2π~)N
. (3.6)
The first is obtained setting Ô → 1̂ in (3.4) and recognizing that the Wigner transform
of the identity operator is 1, whereas the second one can be derived setting Ô → ρ̂n in
(3.4).
3.2 Phase space formulation of the parameter space
geometry
We now use the Wigner function formalism to provide expressions for the quantum metric
tensor and the Berry curvature in both the Abelian and non-Abelian cases. Some years
ago, in reference [64], the author conjectured a phase-space formulation of the Berry
curvature but did not prove it. Moreover, his treatment requires the use of action-angle
variables, and thus, is restricted to the treatment of integrable systems. In contrast
to this, we provide a step-by-step deduction which does not incorporate action-angle
variables. The basic tool is a function of phase space that encodes all the relevant
Phase space formulation of the parameter space geometry 21
information about the parameter space geometry and gives rise to the quantum geometric
tensor. An advantage of our approach is that we can get the quantum metric without
recourse to the wave function, since it only requires the Wigner function of the system
which can be experimentally measured [65].
3.2.1 Abelian case
We first consider the Berry connection for a non-degenerate energy En that corresponds
to the |n〉 state. As we have seen in Section 2.3, this gives rise to an Abelian gauge
structure where the group U(1) underlies the construction. The Berry connection is
given by
A
(n)
i = i〈n|∂in〉. (3.7)
It is easily recognized that it can be rewritten as
A
(n)
i = TrÂ
(n)
i , (3.8)
where the operator Â(n)
i is defined by
Â
(n)
i ≡ i|∂in〉〈n|. (3.9)
This operator is non-Hermitian, and therefore, its Wigner transform
A(n)
i (q, p; x) = i
∞
∫
−∞
dNy e−
ip·y
~
〈
q +
y
2
∣
∣
∣
∣
∂in
〉〈
n
∣
∣
∣
∣
q − y
2
〉
= i
∞
∫
−∞
dNy e−
ip·y
~ ∂iψn
(
q +
y
2
; x
)
ψ∗n
(
q − y
2
; x
)
(3.10)
is a complex function of phase space coordinates. If we take the derivative of equation
(3.3) with respect to the parameter xi, we find that
∂iWn =
1
(2π~)N
∞
∫
−∞
dNy e−
ip·y
~ ∂iψn
(
q +
y
2
; x
)
ψ∗n
(
q − y
2
; x
)
+
1
(2π~)N
∞
∫
−∞
dNy e−
ip·y
~ ψn
(
q +
y
2
; x
)
∂iψ
∗
n
(
q − y
2
; x
)
. (3.11)
Thus, comparing with (3.10), we arrive at
∂iWn =
1
i(2π~)N
A(n)
i − 1
i(2π~)N
A(n)∗
i
=
2
(2π~)N
ImA(n)
i . (3.12)
22 Phase space formulation of the quantum geometric tensor
We can now use (3.2) with Q̂1 → 1̂ and Q̂2 → Â
(n)
i to relate the phase space function
A(n)
i with the Berry connection A(n)
i as
A
(n)
i =
1
(2π~)N
∞
∫
−∞
dNqdNpA(n)
i (q, p; x). (3.13)
This is precisely the expression of the Berry connection in the phase space formalism.
Under the U(1) gauge transformation |n(x)〉 → |n′(x)〉 = eiλn(x)|n(x)〉, the function A(n)
i
transforms as
A(n)
i → A′(n)i = A(n)
i − (2π~)NWn∂iλn, (3.14)
and as a consequence, the Berry connection A(n)
i has the expected gauge transformation
property (2.15)
A
(n)
i → A
′(n)
i = A
(n)
i − ∂iλn. (3.15)
We must point out that only the real part of the function A(n)
i contributes to the Berry
connection A(n)
i . This is straightforward to prove:
A
(n)
i =
1
(2π~)N
∞
∫
−∞
dNqdNpReA(n)
i +
i
(2π~)N
∞
∫
−∞
dNqdNp ImA(n)
i
=
1
(2π~)N
∞
∫
−∞
dNqdNpReA(n)
i +
i
2
∂i


∞
∫
−∞
dNqdNpWn


=
1
(2π~)N
∞
∫
−∞
dNqdNpReA(n)
i , (3.16)
where we used (3.12) and the normalization of the Wigner function (3.5).
Now, we analyze the quantum geometric tensor. We define the operator Ĝ(n)
ij as
Ĝ
(n)
ij ≡
(
Â
(n)†
i − Â
(n)
i
)
Â
(n)
j , (3.17)
and see that the (Abelian) quantum geometric tensor can be obtained as
G
(n)
ij = TrĜ
(n)
ij . (3.18)
From here, if we use the relation Â
(n)†
i − Â
(n)
i = −i∂iρ̂n and in equation (3.2) we set
Q̂1 → −i∂iρ̂n and Q̂2 → Â
(n)
j , we find that
G
(n)
ij = −i
∞
∫
−∞
dNqdNp ∂iWnA(n)
j , (3.19)
Phase space formulation of the parameter space geometry 23
or, using (3.12),
G
(n)
ij = − 2i
(2π~)N
∞
∫
−∞
dNqdNp Im
(
A(n)
i
)
A(n)
j . (3.20)
Thus, we have a formulation of the quantum geometric tensor in terms of the phase
space functions A(n)
i only. Therefore, the A(n)
i encode the relevant information of the
parameter space of a system.
With the quantum geometric tensor at hand, we can extract its real and imaginary
parts to get the quantum metric and the Berry curvature. Hence, the metric tensor takes
the form
g
(n)
ij = ReG
(n)
ij =
2
(2π~)N
∞
∫
−∞
dNqdNp Im
(
A(n)
i
)
Im
(
A(n)
j
)
, (3.21)
or, by virtue of (3.12),
g
(n)
ij =
(2π~)N
2
∞
∫
−∞
dNqdNp ∂iWn∂jWn. (3.22)
On the other hand, the Berry curvature can be written as
F
(n)
ij = −2ImG
(n)
ij =
4
(2π~)N
∞
∫
−∞
dNqdNp Im
(
A(n)
i
)
Re
(
A(n)
j
)
. (3.23)
We want to remark that the phase space formulation to find the quantum metric
tensor (equation (3.22)), does not require the wave functions, it is sufficient to know the
Wigner functions which can actually be obtained from certain functional equations in
phase space [66]. On the other hand, the Berry connection and curvature do require the
knowledge of the wave function.
3.2.2 Non-Abelian case
To treat the non-Abelian case, we define an operator Â(n)
iIJ such that the Wilczek-Zee
connection can be obtained from its trace as
A
(n)
iIJ = TrÂ
(n)
iIJ . (3.24)
The operator turns out to be
Â
(n)
iIJ = i|∂inJ〉〈nI |, (3.25)
and its Wigner transform is
A(n)
iIJ(q, p; x) = i
∞
∫
−∞
dNy e−
ip·y
~ ∂iψnJ
(
q +
y
2
; x
)
ψ∗nI
(
q − y
2
; x
)
. (3.26)
24 Phase space formulation of the quantum geometric tensor
Through the use of (3.24) and (3.2), we get the Wilczek-Zee connection in terms of the
phase space function A(n)
iIJ as
A
(n)
iIJ =
1
(2π~)N
∞
∫
−∞
dNqdNpA(n)
iIJ(q, p; x). (3.27)
This expression is the non-Abelian counterpart of (3.13). The A(n)
iIJ are the entries of a
Hermitian matrix of dimension gn× gn. To see this, we first present the definition of the
non-diagonal Wigner functions WnIJ :
WnIJ =
1
(2π~)N
∞
∫
−∞
dNy e−
ip·y
~ ψnI
(
q +
y
2
; x
)
ψ∗nJ
(
q − y
2
; x
)
. (3.28)
Thus, from equation (3.27), it follows that
A(n)
iIJ = A(n)∗
iJI + i(2π~)N∂iWnJI . (3.29)
Substituting this in (3.27) and using the normalization of the non-diagonal Wigner func-
tions [66],
∞
∫
−∞
dNqdNpWnIJ = δIJ , (3.30)
we find that A(n)
iIJ = A
(n)∗
iJI .
Now, under the SU(N) gauge transformation (2.30), the phase space function A(n)
iIJ
transforms as
A(n)
iIJ → A′(n)iIJ =
gn
∑
K,L=1
[
U
(n)∗
KI A(n)
iKLULJ + i(2π~)NU
(n)∗
KI WnLK∂iU
(n)
LJ
]
, (3.31)
and, as a consequence, the Wilczek-Zee connection has the expected transformation law
(2.38)
A
(n)
iIJ → A
′(n)
iIJ =
gn
∑
K,L=1
U
(n)∗
KI A
(n)
iKLU
(n)
LI + i
gn
∑
K=1
U
(n)∗
KI ∂iU
(n)
KJ . (3.32)
Now, the non-Abelian quantum geometric tensor (2.29) can be written as
G
(n)
ijIJ = TrĜ
(n)
ijIJ , (3.33)
where the operator Ĝ(n)
ijIJ is defined as
Ĝ
(n)
ijIJ ≡ −i∂iP̂ nÂ
(n)
jIJ , (3.34)
Illustrative examples 25
and P̂ n =
∑gn
I=1 |nI〉〈nI | is the projection operator onto the nth subspace. Setting
Q̂1 → −i∂iP̂ n and Q̂2 → Â
(n)
jIJ in (3.2), we find
G
(n)
ijIJ = −i
∞
∫
−∞
dNqdNp
gn
∑
K=1
∂iWnKKA(n)
jIJ . (3.35)
To cast this expression entirely in terms of A(n)
iIJ , we use the relation
ImA(n)
iKK =
(2π~)N
2
∂iWnKK , (3.36)
which stems from (3.26) and (3.28). Therefore, substituting (3.36) in (3.35), we arrive
at the non-Abelian quantum geometric tensor:
G
(n)
ijIJ = − 2i
(2π~)N
∞
∫
−∞
dNqdNp
gn
∑
K=1
Im
(
A(n)
iKK
)
A(n)
jIJ . (3.37)
Now, we can extract the non-Abelian quantum metric tensor using (2.32) and get
g
(n)
ijIJ =
−i
(2π~)N
∞
∫
−∞
dNqdNp
gn
∑
K=1
Im
(
A(n)
iKK
)(
A(n)
jIJ −A(n)∗
jJI
)
, (3.38)
or, we can cast it entirely in terms of Wigner functions through (3.29) and (3.36) as
g
(n)
ijIJ =
(2π~)N
2
∞
∫
−∞
dNqdNp
gn
∑
K=1
∂iWnKK∂JWnIJ . (3.39)
On the other hand, the Wilczek-Zee curvature is obtained from (3.37) using (2.33), and
it takes the form
F
(n)
ijIJ =
2
(2π~)N
∞
∫
−∞
dNqdNp
gn
∑
K=1
Im
(
A(n)
iKK
)(
A(n)
jIJ +A(n)∗
jJI
)
. (3.40)
In this way, we have formulated the basic geometric structures that underlie the param-
eter space in both the non-degenerate and degenerate systems.
3.3 Illustrative examples
In this section, our aim is to present two simple systems that can be treated with the
phase space formulation. First, we analyze the generalized harmonic oscillator whose
geometric tensor has an Abelian geometrical structure, and then, we study a set of three
coupled harmonic oscillators that has a degenerate spectrum, and thus, gives rise to a
non-Abelian quantum geometric tensor.
26 Phase space formulation of the quantum geometric tensor
3.3.1 Generalized harmonic oscillator
The Hamiltonian of the generalized harmonic oscillator is
Ĥ =
1
2
[
Xq̂2 + Y (q̂p̂+ p̂q̂) + Zp̂2
]
, (3.41)
where the adiabatic parameters are x = {xi} = (X, Y, Z) with i = 1, 2, 3. The Schrödinger
equation takes the form
− Z~2
2
d2ψn
dq2
− i~Y q
dψn
dq
+
(
Xq2
2
− i~
Y
2
)
ψn = Enψn, (3.42)
and the resulting wave function is
ψn(q; x) =
( ω
Z~
)1/4
χn
(
q
√
ω
Z~
)
exp
(
− iY q
2
2Z~
)
, (3.43)
where ω =
√
XZ − Y 2 is the frequency, so this system has an oscillatory behavior as
long as XZ − Y 2 > 0. Moreover, χn(ξ) are the Hermite functions
χn(ξ) =
(
2nn!
√
π
)−1/2
e−ξ
2/2Hn(ξ), (3.44)
with Hn(ξ) the nth Hermite polynomial. The energy is restricted to take the values
En = (n+ 1/2)~ω (n = 0, 1, 2, ...), and of course, no degeneracy exists.
The Wigner function of the generalized harmonic oscillator can be obtained from
that of the simple harmonic oscillator through the transformation
Q =
q√
Z
, P =
√
Z
(
p+
Y
Z
q
)
, (3.45)
which casts the associated classical Hamiltonian H = 1
2
(Xq2 + 2Y qp + Zp2) into the
standard form
H =
1
2
(P 2 + ω2Q2). (3.46)
Thus, the Wigner function is [66]
Wn(q, p; x) =
(−1)n
π~
e−λ/2Ln(λ), (3.47)
where λ = 4H/(~ω) and Ln(λ) are the Laguerre polynomials.
We now compute the phase space function A(n)
i using equation (3.10). Through the
use of the following properties of the Hermite functions,
dχn
dξ
=
√
n
2
χn−1 −
√
n+ 1
2
χn+1,
ξχn =
√
n
2
χn−1 +
√
n+ 1
2
χn+1, (3.48)
Illustrative examples 27
we arrive at
A(n)
i =
π~Z
2ω
[
i∂i
(ω
Z
)
Ξ(−)
n + ∂i
(
Y
Z
)
(
Ξ(+)
n + (2n+ 1)Wn
)
]
. (3.49)
Here, Wn is the Wigner function (3.47), and the Ξ
(±)
n are
Ξ(±)
n =
√
n(n− 1)Wn−2,n ±
√
(n+ 1)(n+ 2)Wn+2,n, (3.50)
where
Wn±2,n =
1
2π~
∞
∫
−∞
dy e−
ipy
~ ψn±2
(
q +
y
2
; x
)
ψ∗n
(
q − y
2
; x
)
, n ≥ 2 (3.51)
are the non-diagonal Wigner functions [66]
Wn−2,n =
(−1)n+12(P − iωQ)2
√
n(n− 1)π~2ω
e−
λ
2L
(2)
n−2(λ),
Wn+2,n =
(−1)n+12(P + iωQ)2
√
(n+ 1)(n+ 2)π~2ω
e−
λ
2L(2)
n (λ), (3.52)
with L
(α)
n (λ) the associated Laguerre polynomials. Substituting (3.49) in (3.13) and
using the orthogonality of the non-diagonal Wigner functions,
∞
∫
−∞
dqdpWn±2,n = 0, (3.53)
we find the Berry connection A(n) = A
(n)
i dxi, whose three components are
A
(n)
1 = 0, A
(n)
2 =
(
n+
1
2
)
1
2ω
, A
(n)
3 = −
(
n+
1
2
)
Y
2Zω
. (3.54)
Moreover, the components of the Berry curvature are found as F (n)
ij = ∂iA
(n)
j − ∂jA
(n)
i ,
resulting in
F
(n)
23 = −
(
n+
1
2
)
X
4ω3
, F
(n)
31 = −
(
n+
1
2
)
Y
4ω3
, F
(n)
12 = −
(
n+
1
2
)
Z
4ω3
. (3.55)
We finally find the quantum metric tensor. The derivative of the Wigner function
(3.47) with respect to the parameter xi is
∂iWn = −(−1)n
2π~
e−λ/2
[
Ln(λ)− 2
dLn(λ)
dλ
]
∂iλ, (3.56)
28 Phase space formulation of the quantum geometric tensor
thus, substituting it in (3.22) we find that
g
(n)
ij =
2π~
2
∞
∫
−∞
dqdp ∂iWn∂jWn
=
1
4π~
∞
∫
−∞
dqdp e−
λ
2
(
Ln − 2
dLn
dλ
)2
∂iλ∂jλ. (3.57)
Taking the required partial derivatives, and then, applying the transformation
q =
√
~λZ
2ω
sinφ, p =
√
~λZ
2ω
(
−Y
Z
sinφ+
ω
Z
cosφ
)
, (3.58)
and the fact that
∞
∫
−∞
dq
∞
∫
−∞
dp =
~
4
2π
∫
0
dφ
∞
∫
0
dλ, (3.59)
we can recast the integral in terms of the variables (φ, λ). The integral in φ is easily
carried out, and for the integral in λ we need to use the relation
∞
∫
0
dλλ2e−λ
(
Ln − 2
dLn
dλ
)2
= 2(n2 + n+ 1), (3.60)
after which we find the quantum metric tensor:
g
(n)
ij =
n2 + n+ 1
32ω4


Z2 −2Y Z 2Y 2 −XZ
−2Y Z 4XZ −2XY
2Y 2 −XZ −2XY X2

 . (3.61)
3.3.2 Three coupled harmonic oscillators
We now analyze a system of coupled harmonic oscillators which possesses a non-Abelian
parameter space structure for excited states. Our aim here is to compute the non-Abelian
quantum metric tensor.
Let us consider the Hamiltonian
Ĥ =
1
2
3
∑
a=1
(p̂2a + kq̂2a) +
k′
2
[
(q̂1 − q̂2)
2 + (q̂2 − q̂3)
2 + (q̂3 − q̂1)
2
]
, (3.62)
where the parameters are x = {xi} = (k, k′), i = 1, 2. We begin by uncoupling the
oscillators by means of the transformation


Q̂1
Q̂2
Q̂3

 =



1√
3
1√
3
1√
3
− 1√
2
0 1√
2
1√
6
−
√
2
3
1√
6





q̂1
q̂2
q̂3

 , (3.63)
Illustrative examples 29
and similarly for the momenta. The resulting Hamiltonian reads
Ĥ =
1
2
3
∑
a=1
(P̂ 2
a + ω2
aQ̂
2
a), (3.64)
where the frequencies are ω1 =
√
k, ω2 = ω3 =
√
k + 3k′. The system is then separable,
and thus, the wave function is
ψn1,n2,n3(q1, q2, q3; x) = ψn1(Q1; x)ψn2(Q2; x)ψn3(Q3; x), (3.65)
with na = 1, 2, .... The individual wave functions ψa have the form
ψa(Qa; x) =
(ωa
π~
)1/4 1√
2nana!
Hna
(
√
ωa
~
Qa
)
exp
(
−ωa
2~
Q2
a
)
, (3.66)
where Hna
(ξ) are the Hermite polynomials, and the energy takes the values
En1,n2,n3 =
(
n1 +
1
2
)
~ω1 + (n2 + n3 + 1)~ω2. (3.67)
To compute the non-Abelian quantum metric tensor, we choose the first excited-state
wave functions ψ0,0,1 and ψ0,1,0 which have the same energy E1 =
1
2
~ω1+2~ω2, and there-
fore, constitute a degenerate set with g1 = 2. Using the notation {ψ(1)I} = (ψ0,0,1, ψ0,1,0)
with I, J = 1, 2, we obtain the non-diagonal Wigner functions using equation (3.28):
W(1)IJ =
1
(2π~)3
∞
∫
−∞
d3y e−
ip·y
~ ψ(1)I
(
q +
y
2
; x
)
ψ∗(1)J
(
q − y
2
; x
)
. (3.68)
At this point, it is useful to make the change of integration variables y → Y as


Y1
Y2
Y3

 =



1√
3
1√
3
1√
3
− 1√
2
0 1√
2
1√
6
−
√
2
3
1√
6





y1
y2
y3

 , (3.69)
so that the integral can be separated into three independent pieces which are easily
performed. The resulting non-diagonal Wigner functions are
W(1)11 =
1
(π~)3
(λ3 − 1)e−
λ1+λ2+λ3
2 ,
W(1)12 =
2
π3~4ω2
(P2 − iω2Q2)(P3 + iω2Q3)e
−λ1+λ2+λ3
2 ,
W(1)21 =
2
π3~4ω2
(P2 + iω2Q2)(P3 − iω2Q3)e
−λ1+λ2+λ3
2 ,
W(1)22 =
1
(π~)3
(λ2 − 1)e−
λ1+λ2+λ3
2 , (3.70)
30 Phase space formulation of the quantum geometric tensor
where λa = 4Ha/(~ωa) and Ha = (P 2
a +ω
2
aQ
2
a)/2. We substitute these functions in (3.39)
and find the non-Abelian quantum geometric tensor, whose components are
g
(1)
ij11(x) = g
(1)
ij22(x) =
1
32
(
1
ω4
1
+ 4
ω4
2
12
ω4
2
12
ω4
2
36
ω4
2
)
,
g
(1)
ij12(x) = g
(1)
ij21(x) =
(
0 0
0 0
)
. (3.71)
Chapter 4
Classical analog of the quantum
geometric tensor
We devote this chapter to the description of the classical counterparts of the geomet-
ric objects that appeared in the quantum realm. First, we review the Hannay angle,
connection, and curvature. Then, we introduce one of the most important results of
this work: the notion of the classical generator of translations in parameter space which
will allow us to find the classical analog of the metric tensor. We corroborate that our
proposed classical metric tensor satisfies the desired properties and then obtain it from
the semiclassical approximation of the path integral approach introduced in Chapter 2.
After this, we prove the equivalence of the generator and time-dependent approaches
which parallels that of their quantum counterpart, and finally, we provide the differ-
ent expressions to compute the Hannay curvature and the classical metric tensor. The
results of this chapter have been published in references [67] and [68].
4.1 The Hannay angle, connection and curvature
To formulate the adiabatic theorem in classical mechanics, we require an integrable
system with a Hamiltonian H(q, p; x) that depends on the parameters x = {xi}, i =
1, ...,M , where q = {qa} and p = {pa}, a = 1, ..., N are the coordinates and their
canonical momenta, respectively. Since we have an integrable system, we can introduce
action-angle variables I = {Ia} and φ = {φa} that satisfy Hamilton’s equations with
respect to the transformed Hamiltonian [69]
K(φ, I; x) = H(I; x) +
∂S(α)(q, I; x)
∂t
= H(I; x) + (∂iS
(α))q,I ẋ
i, (4.1)
where H(I; x) = H(q(φ, I; x), p(φ, I; x); x) and S(α)(q, I; x) is the generating function of
the canonical transformation (q, p) → (φ, I). The subscript indicates that the derivative
is taken with respect to the parameters leaving (q, I) fixed, and the superscript α labels
different branches of the multivalued generating function. Also, we have used the Einstein
31
32 Classical analog of the quantum geometric tensor
summation convention over the index i. Using the chain rule and selecting the branch
0 ≤ φ < 2π, we can easily see that
(∂iS)q,I = −Gi, (4.2)
with
Gi(φ, I; x) ≡ pa(∂iq
a)φ,I − (∂iS)φ,I . (4.3)
Here, too, we are using the Einstein summation convention over a. The adiabatic theorem
states that the action variables Ia remain invariant during the slow evolution of the
parameters and that the angle variables change while the system is taken through a
closed curve in parameter space is
∆φa =
T
∫
0
dt′ ωa(I; x)− ∂
∂Ia
∮
C
dxi 〈Gi(φ, I; x)〉, (4.4)
where ωa(I; x) = ∂H(I; x)/∂Ia and
〈f(φ, I; x)〉 = 1
(2π)N
2π
∫
0
dNφ f(φ, I; x) (4.5)
is the torus average of the function f . The first term in (4.4) represents a dynamical
change, while the second is a geometric change called the Hannay angle [8] and denoted
as ∆φaH . In analogy to the quantum case, we introduce the one-form
A = Aidx
i = 〈Gi〉dxi, (4.6)
so that using Stokes theorem, the ath Hannay angle can be written as
∆φaH(C) = − ∂
∂Ia
∫
Σ
F, (4.7)
where F is a two-form such that F = dA and C = ∂Σ. In terms of components,
F =
1
2
Fijdx
i ∧ dxj, (4.8)
with
Fij = ∂iAj − ∂jAi = 〈(∂ipa)φ,I(∂jqa)φ,I − (∂jpa)φ,I(∂iq
a)φ,I〉. (4.9)
Now, consider the infinitesimal canonical transformation
qa(x) → qa(x+ δx) = qa(x) + δqa, (4.10)
pa(x) → pa(x+ δx) = pa(x) + δpa, (4.11)
The Hannay angle, connection and curvature 33
where
δqa ≡ qa(x′)− qa(x) = (∂iq
a)φ,Iδx
i, (4.12)
δpa ≡ pa(x
′)− pa(x) = (∂ipa)φ,Iδx
i, (4.13)
with x′ = x + δx. Note that the variation δqa leaves the action-angle variables fixed,
and can be written in terms of the more familiar virtual and total variations, δ̃qa =
q′a(x)− qa(x) and δqa = q′a(x′)− qa(x), respectively, as
δqa = δqa − δ̃qa. (4.14)
We set out to prove the crucial result that the Gi are the generators of this infinitesimal
canonical transformation, i.e.,
(∂iq
a)φ,I = {qa, Gi}q,p =
∂Gi
∂pa
, (4.15)
(∂ipa)φ,I = {pa, Gi}q,p = −∂Gi
∂qa
. (4.16)
In the first place, we differentiate the defining relation of a canonical transformation,
paδ̃q
a − Iaδ̃φ
a = δ̃f, (4.17)
with respect to xi leaving (φ, I) fixed. Here, f = S(q, I; x) − φaIa, and the virtual
variation δ̃ is taken at fixed time (frozen parameters). We get
(∂ipa)φ,I δ̃q
a + paδ̃(∂iq
a)φ,I = δ̃(∂iS)φ,I , (4.18)
where we used the fact that δ̃ commutes with ∂i. Next, we perform the virtual variation
of (4.3) which yields
δ̃Gi = δ̃pa(∂iq
a)φ,I + paδ̃(∂iq
a)φ,I − δ̃(∂iS)φ,I . (4.19)
Now, we combine this equation with (4.18), resulting in
δ̃Gi = −(∂ipa)φ,I δ̃q
a + (∂iq
a)φ,I δ̃pa. (4.20)
On the other hand, if we consider Gi as a function of (q, p), it follows that
δ̃Gi =
∂Gi
∂qa
δ̃qa +
∂Gi
∂pa
δ̃pa. (4.21)
Comparing these two previous equations, we find (4.15) and (4.16). Therefore, since the
Gi generate translations in parameter space, they are the classical analogs of the quantum
operators Ĝi. Furthermore, using (4.15) and (4.16), we can write the components of the
two-form F (4.9) as
Fij = −〈{Gi, Gj}〉. (4.22)
34 Classical analog of the quantum geometric tensor
Notice that written in this form, we can clearly see that Fij is the classical analog of
the Berry curvature (2.46), and that the torus average 〈·〉 is the classical analog of the
expectation value 〈·〉n.
Finally, we address the issue of gauge transformations. In this case, the analog of
the quantum local phase transformation (2.19) is a parameter-dependent angle shift [9]
given by the canonical transformation
I ′a = Ia, φ′a = φa +
∂λ(I; x)
∂Ia
, (4.23)
which has the generating function [69]
F2(φ, I
′; x) = φaI ′a + λ(I ′; x). (4.24)
Note that this angle variable shift may depend on I, which is analogous to having a
different Berry phase for each n. The transformation from (q, p) to (φ, I) gives the
Hamiltonian (4.1)
K(φ, I; x) = H(I; x)−Gi(φ, I; x)ẋ
i. (4.25)
Then, the transformation (φ, I) → (φ′, I ′) yields
K ′(φ′, I ′; x) = K(φ′, I ′; x) +
(
∂F2(φ, I
′; x)
∂t
)
φ=φ(φ′,I′;x)
, (4.26)
where K(φ′, I ′; x) = K(φ(φ′, I ′; x), I ′; x). Hence, combining both Hamiltonians, we get
K ′(φ′, I ′; x) = H −
[
Gi −
(
∂F2
∂xi
)]
ẋi. (4.27)
This expression defines the generator G′(φ′, I ′; x) as
G′(φ′, I ′; x) = Gi −
(
∂F2
∂xi
)
. (4.28)
Applying the chain rule on (4.24), we find
G′i(φ
′, I ′; x) = Gi − ∂iλ. (4.29)
The new components of the one-form are then
A′i(I
′; x) = 〈G′i〉′, (4.30)
where G′i = G′i(φ
′, I ′; x) and 〈·〉′ indicates the torus average over the angle variable φ′.
Using (4.29) we can see that
〈G′i〉′ =
1
(2π)N
2π
∫
0
dNφ′ [Gi(φ
′, I ′; x)− (∂iλ(I
′; x))I′ ]
=
1
(2π)N
2π−b1
∫
−b1
dφ1 · · ·
2π−bN
∫
−bN
dφN [Gi(φ, I; x)− (∂iλ(I; x))I ]
=〈Gi〉 − ∂iλ, (4.31)
Classical analog of the quantum metric tensor 35
where we performed the change of angle variables φ′a → φa, defined ba = ∂λ(I; x)/∂Ia,
and used the periodicity of pa, (∂iqa)φ,I and (∂iS)φ,I in each angle variable which implies
the periodicity of Gi. The function (∂iS)φ,I is seen to be periodic by writing
S(q, I; x) =
N
∑
a=1
Sa(qa, I; x). (4.32)
If we use the definition of the action variable Ia,
Sa(φ+ 2π, I; x)− Sa(φ, I; x) = 2πIa, (4.33)
where Sa(φ, I; x) = Sa(q(φ, I; x), I; x), and differentiate with respect to xi, the periodicity
(∂iS)φ,I is evident. We have found that the one-form components Ai transform as
Ai → A′i = Ai − ∂iλ, (4.34)
so the one-form A defines an Abelian connection [9]. On the other hand, F = dA is the
curvature associated with this connection and it is left gauge invariant, since
F ′ = dA′ = dA− d2λ = dA. (4.35)
4.2 Classical analog of the quantum metric tensor
Now that we have introduced the classical generators Gi, and in complete analogy
with the quantum case (2.49), it is natural to define the distance between the points
(q(x), p(x)) and (q(x) + δq, p(x) + δp) of phase space as
ds2 = 〈(∆G)2〉, (4.36)
where ∆G = ∆Giδx
i, with ∆Gi = Gi − 〈Gi〉. Hence, expanding (4.36), we find that the
distance takes the form
ds2 = (〈GiGj〉 − 〈Gi〉〈Gj〉) δxiδxj. (4.37)
Therefore, we define the classical analog of the quantum metric tensor as
gij(I; x) = 〈GiGj〉 − 〈Gi〉〈Gj〉. (4.38)
From now on, we will refer to it as classical metric.
We can easily verify that our classical metric: (i) transforms as a tensor under the
change of parameters, (ii) it is gauge invariant, and (iii) it is positive semidefinite. To
prove the first property, we note from (4.3) that given the transformation xi → x′i =
x′i(x), the generator Gi changes as
Gi(φ, I; x) → G′i(φ, I; x
′) =
∂xj
∂x′i
Gj(φ, I; x(x
′)). (4.39)
36 Classical analog of the quantum geometric tensor
This result, together with the definition (4.38) leads to
g′ij(I; x
′) =
∂xk
∂x′i
∂xl
∂x′j
gkl(I; x). (4.40)
The proof of property (ii) is readily carried out by considering the transformations (4.29)
and (4.31). We find that the first term of the classical metric transforms as
〈G′iG′j〉′ = 〈GiGj〉 − 〈Gi〉(∂jλ)− 〈Gj〉(∂iλ) + (∂iλ)(∂jλ). (4.41)
Despite this term being a symmetric tensor, it should not be considered as a classical
metric candidate, since it is clearly not gauge invariant. We see that, analogously to the
Provost quantum metric (2.20), the second term 〈Gi〉〈Gj〉 compensates this change and
we are left with
g′ij(I
′; x) = 〈GiGj〉 − 〈Gi〉〈Gj〉 = gij(I; x). (4.42)
Regarding the third property, it is easy to see from expression (4.36) that dL2 ≥ 0
since the variance (∆G)2 is non-negative. Thus, our classical metric (4.38) is positive
semidefinite. Moreover, we observe that the classical analog of the quantum persistence
of the system in the state |n〉, is the invariance of the action variable I. Finally, note
that from (4.15) we can see that
{qa,∆Gi} = (∂i − Ai)q
a, (4.43)
which, in an analogous manner to the quantum case (cf. (2.50)), resembles the structure
of a covariant derivative Di = ∂i − Ai with connection Ai.
4.2.1 Behavior under canonical transformations
So far, we have seen how the classical metric tensor behaves under parameter transfor-
mations xi → x′i = x′i(x), however, nothing has been said about the transformation
properties of the generators and the classical metric under canonical transformations. In
some problems, it may be easier to work with a special set of phase space coordinates,
thus, the rule for the transformation of generators constitutes a crucial result of this
work. We must say that the quantum counterpart of this problem was not considered,
although it might be developed through the use of unitary transformations, at least those
that correspond to linear canonical transformations [70].
As we know, the generators of displacements in the parameters of phase space coor-
dinates (q, p) are (4.2)
Gi = −
(
∂S
∂xi
)
q,I
, (4.44)
where the function S(q, I; x) effects the transformation from the original variables (q, p)
to the action-angle variables (φ, I). Now, consider the canonical transformation (q, p) →
(Q,P ) performed by a type-2 generating function F = F (q, P ; x). We might as well
Classical analog of the quantum metric tensor 37
introduce the generator of displacements in the parameters of the variables (Q,P ), which
are given by
G̃i = −
(
∂S̃
∂xi
)
Q,I
, (4.45)
where S̃(Q, I; x) is the generating function of the transformation (Q,P ) → (φ, I). Our
goal is to find the relation between the functions Gi and G̃i. Given that there are
three transformations involved in the process, we need to know the rule for composing
canonical transformations through their generating functions. It turns out that the
resulting generating function S is [71, 72]
S(q, I; x) = S̃(Q, I; x) + F (q, P ; x)−QaPa. (4.46)
For our purposes, it is convenient to cast this composition rule as a function of the
variables (Q, I; x) as follows:
S̃(Q, I; x) = S(q(Q, I; x), I; x)− F (q(Q, I; x), P (Q, I; x); x) +QaPa(Q, I; x). (4.47)
We take the derivative of this expression with respect to xi with the aid of the chain rule
and find that
(
∂S̃
∂xi
)
Q,I
=
(
∂S
∂qa
)
I,x
(
∂qa
∂xi
)
Q,I
+
(
∂S
∂xi
)
q,I
−
[(
∂F
∂qa
)
P,x
(
∂qa
∂xi
)
Q,I
+
(
∂F
∂Pa
)
q,x
(
∂Pa
∂xi
)
Q,I
+
(
∂F
∂xi
)
q,P
]
+Qa
(
∂Pa
∂xi
)
Q,I
. (4.48)
Using the following equations of a type-2 canonical transformation
pa =
(
∂S
∂qa
)
I,x
, pa =
(
∂F
∂qa
)
P,x
, Qa =
(
∂F
∂Pa
)
q,x
, (4.49)
we arrive at
(
∂S̃
∂xi
)
Q,I
=pa
(
∂qa
∂xi
)
Q,I
+
(
∂S
∂xi
)
q,I
−
[
pa
(
∂qa
∂xi
)
Q,I
+Qa
(
∂Pa
∂xi
)
Q,I
+
(
∂F
∂xi
)
q,P
]
+Qa
(
∂Pa
∂xi
)
Q,I
, (4.50)
which simplifies to
(
∂S̃
∂xi
)
Q,I
=
(
∂S
∂xi
)
q,I
−
(
∂F
∂xi
)
q,P
. (4.51)
Identifying the generators Gi and G̃i through (4.44) and (4.45), we arrive at the desired
formula
Gi = G̃i −
(
∂F
∂xi
)
q,P
. (4.52)
38 Classical analog of the quantum geometric tensor
This apparently simple relation contains a powerful result: if the generating function
of the transformation (q, p) → (Q,P ) does not depend on the parameters x, then the
classical metric resulting from Gi is the same as that obtained with G̃i. Thus, if the
description of the system in question is simpler in terms of the coordinates (Q,P ) rather
than in coordinates (q, p), and the type-2 generating function of the (q, p) → (Q,P )
transformation does not depend on parameters, then the metric can be found through
the formula gij = 〈G̃iG̃j〉 − 〈G̃i〉〈G̃j〉. It is worth noting the inhomogeneous charac-
ter of equation (4.52), as opposed to the homogeneous tensor transformation rule for
parameters (4.39).
4.3 Time-dependent approach to the classical metric
and curvature
In this section, we introduce a time-dependent approach to the classical metric and the
Hannay curvature [68]. This formulation is the classical analog of the path integral
approach of Section 2.5, and actually stems from the semiclassical approximation of
equations (2.56) and (2.57), as we will show next.
Let us recall the expressions for the quantum metric tensor and the Berry curvature:
g
(0)
ij (x) = − 1
~2
0
∫
−∞
dt1
∞
∫
0
dt2
(
1
2
〈[Ôi(t1)Ôj(t2)]+〉0 − 〈Ôi(t1)〉0〈Ôj(t2)〉0
)
, (4.53)
F
(0)
ij (x) =
1
i~2
0
∫
−∞
dt1
∞
∫
0
dt2 〈[Ôi(t1), Ôj(t2)]−〉0, (4.54)
where [·, ·]+ stands for the anticommutator and [·, ·]− for the commutator. We assume
that the Heisenberg equations of motion for the N position and N momentum operators,
dq̂a
dt
= − i
~
[q̂a, Ĥ],
dp̂a
dt
= − i
~
[p̂a, Ĥ], , a = 1, ..., N (4.55)
can be integrated, so that q̂(t) = {q̂a(t)} and p̂(t) = {p̂a(t)} can be expressed as functions
of the initial (Schrödinger) operators q̂ = q̂(t = 0) and p̂ = p̂(t = 0), and time. In this
way, the Hamiltonian deformation Ôi(t) will read
Ôi(t) = Ôi(q̂(q̂0, p̂0, t; x), p̂(q̂0, p̂0, t; x), t; x). (4.56)
Thus, the expectation values appearing in (4.53) and (4.54) can be written as
〈Ôi(t)〉0 = 〈ψ0(x)|Ôi(t)|ψ0(x)〉
=
∫
dq0 ψ
∗
0(q0; x)Oi
(
q0,−i~
∂
∂q0
, t; x
)
ψ0(q0; x), (4.57)
Time-dependent approach to the classical metric and curvature 39
and as
〈[Ôi(t1)Ôj(t2)]±〉0 = 〈ψ0(x)|[Ôi(t1), Ôj(t2)]±|ψ0(x)〉
=
∫
dNq0 ψ
∗
0(q0; x)
×
[
Oi
(
q0,−i~
∂
∂q0
, t1; x
)
,Oj
(
q0,−i~
∂
∂q0
, t2; x
)]
±
ψ0(q0; x).
(4.58)
In the spirit of Berry’s work [9], we use the semiclassical approximation of the wave
function for an integrable system with N degrees of freedom [73, 74]:
ψ0(q0; x) ≃
∑
α
a(α)(q0, I0; x) exp
[
i
~
S(α)(q0, I0; x)
]
, (4.59)
where
(
a(α)(q0, I0; x)
)2
=
1
(2π)N
det
[
∂φa0(α)(q0, I0; x)
∂qb0
]
, (4.60)
and α denotes the branch of the multi-valued function S(α)(q0, I0; x) which generates
the transformation (q0, p0) → (φ0, I0). Since the action I0 remains constant, we may set
I0 = I. Substituting (4.59) in (4.57) and (4.58), and taking into account that terms
coming from different branches are suppressed due to a strongly oscillating behavior, we
arrive at
〈Ôi(t)〉0 =
∫
dNq0
(2π)N
∑
α
det
[
∂φa0(α)(q0, I; x)
∂qb0
]
Oi
(
q0,
∂S(α)
∂q0
, t; x
)
+O(~), (4.61)
and
〈[Ôi(t1)Ôj(t2)]±〉0 =
∫
dNq0
(2π)N
∑
α
det
[
∂φa0(α)(q0, I; x)
∂qb0
]
×
[
Oi
(
q0,
∂S(α)
∂q0
, t1; x
)
,Oj
(
q0,
∂S(α)
∂q0
, t2; x
)]
±
+O(~). (4.62)
We replace ∂S(α)
∂q0
by p(α)0 , following the transformation equations of the canonical trans-
formation to action-angle variables. For bosonic operators, we must replace the anticom-
mutators by products of functions, and the commutators by i~ times the non-equal-time
Poisson brackets
{Oi(t1),Oj(t2)} =
N
∑
a=1
(
∂Oi(t1)
∂qa0
∂Oj(t2)
∂p
(α)
0a
− ∂Oi(t1)
∂p
(α)
0a
∂Oj(t2)
∂qa0
)
. (4.63)
40 Classical analog of the quantum geometric tensor
If, on the other hand, we are dealing with fermionic operators, we replace the anticom-
mutators by Poisson brackets, and the commutators by products. Assuming the bosonic
case and neglecting terms of order O(~), we have that
〈Ôi(t)〉0 ≃
∫
dNq0
(2π)N
∑
α
det
[
∂φa0(α)(q0, I; x)
∂qb0
]
Oi
(
q0, p
(α)
0 , t; x
)
, (4.64)
〈[Ôi(t1)Ôj(t2)]+〉0 ≃ 2
∫
dNq0
(2π)N
∑
α
det
[
∂φa0(α)(q0, I; x)
∂qb0
]
×Oi
(
q0, p
(α)
0 , t1; x
)
Oj
(
q0, p
(α)
0 , t2; x
)
, (4.65)
and
〈[Ôi(t1)Ôj(t2)]−〉0 ≃ i~
∫
dNq0
(2π)N
∑
α
det
[
∂φa0(α)(q0, I; x)
∂qb0
]
×
{
Oi
(
q0, p
(α)
0 , t1; x
)
,Oj
(
q0, p
(α)
0 , t2; x
)
}
. (4.66)
We now choose the branch defined by 0 ≤ φ0 < 2π which makes φ0(α), and therefore,
p
(α)
0 single-valued. In addition to this, we effect the change to the angle variable in the
integrals. Thus,
〈Ôi(t)〉0 ≃
1
(2π)N
∫
dNφ0 Oi(q0, p0, t; x), (4.67)
〈[Ôi(t1)Ôj(t2)]+〉0 ≃
2
(2π)N
∫
dNφ0 Oi(q0, p0, t1; x)Oj(q0, p0, t2; x), (4.68)
〈[Ôi(t1)Ôj(t2)]−〉0 ≃ i~
1
(2π)N
∫
dNφ0 {Oi(q0, p0, t1; x),Oj(q0, p0, t2; x)}. (4.69)
Denoting by 〈·〉0 the torus average of over the initial angle variables 1
(2π)N
∫
dNφ0 (·), we
can cast the above equations as
〈Ôi(t)〉0 ≃ 〈Oi(t)〉0, (4.70)
〈[Ôi(t1)Ôj(t2)]+〉0 ≃ 2〈Oi(t1)Oj(t2)〉0, (4.71)
〈[Ôi(t1)Ôj(t2)]−〉0 ≃ i~〈{Oi(t1),Oj(t2)}〉0, (4.72)
where Oi(t) = Oi(q0, p0, t; x) are the deformation functions
Oi =
(
∂H
∂xi
)
q,p
(4.73)
Equivalence between the two approaches 41
obtained solving Hamilton’s equations and then expressing the solutions in terms of
initial conditions (q0, p0) and time, which in turn, must be written in terms of initial
action-angle variables (φ0, I) and time.
Plugging (4.70) and (4.71) into (4.53), we arrive at
g
(0)
ij (x) ≃ 1
~2
gij(I; x), (4.74)
where
gij(I; x) = −
0
∫
−∞
dt1
∞
∫
0
dt2
(
〈Oi(t1)Oj(t2)〉0 − 〈Oi(t1)〉0〈Oj(t2)〉0
)
(4.75)
is the classical metric. Substituting (4.70) and (4.72) in (4.54), we find
F
(0)
ij (x) ≃ 1
~
Fij(I; x), (4.76)
where
Fij(I; x) =
0
∫
−∞
dt1
∞
∫
0
dt2 〈{Oi(t1),Oj(t2)}〉0, (4.77)
is the Hannay curvature. In these expressions, it is understood that the action variable I
is quantized according to the corrected Bohr-Sommerfeld rule, also called the Einstein-
Brillouin-Keller (EBK) rule [74]
Ia =
(
na +
µa
4
)
~, (4.78)
where µa are the Maslov indices [73]. Previous works [9, 75] had already obtained the
relation (4.76), however, neither of them tackled the classical analog of the quantum
metric tensor.
4.4 Equivalence between the two approaches
In this section, we prove a crucial result of our work: the equivalence between the time-
dependent approach to the classical metric and the Hannay curvature, and the generator
approach. This justifies our initial proposal of the classical metric (4.38).
Consider an integrable system of N degrees of freedom with Hamiltonian H(q, p; x),
where q = {qa} and p = {pa} with a = 1, ..., N are the phase space coordinates and
x = {xi} with i = 1, ...,M is a set of M adiabatic parameters. The classical deformation
functions
Oi =
(
∂H
∂xi
)
q,p
(4.79)
42 Classical analog of the quantum geometric tensor
are functions of q, p and the parameters x. In the periodic motion regime, action-angle
variables (φ, I) can be introduced and any function of q and p will be itself periodic [76],
thus, it can be expanded in Fourier series in terms of the angle variables φ. Using the
compact notation Oi(t) ≡ Oi(q(t), p(t); x), we have that
Oi(t) =
∑
n
β(i)
n e
in·(φ0+ωt), (4.80)
where φ0 = (φ1
0, ...φ
N
0 ) are the initial angle variables, ω = (ω1, ..., ωN) are the frequencies
derived as ωa = ∂H/∂Ia, and the β(i)
n are the Fourier coefficients of the time-independent
deformation functions Oi ≡ Oi(q0, p0; x), i.e.,
β(i)
n ≡ β(i)
n (I; x) =
1
(2π)N
2π
∫
0
dNφ0 Oie
−in·φ0 , (4.81)
where n = (n1, ..., nN), and every na runs over all the integers.
4.4.1 Classical metric
We begin with the classical metric (4.75). We first consider the torus average of the
Oi(t):
〈Oi(t)〉0 =
1
(2π)N
2π
∫
0
dNφ0
∑
n
β(i)
n e
in·(φ0+ωt)
=
∑
n
β(i)
n e
in·ωt


1
(2π)N
2π
∫
0
dNφ0 e
in·φ0


=
∑
n
β(i)
n e
in·ωtδn,0 = β
(i)
0 . (4.82)
From equation (4.81), we conclude that
〈Oi(t)〉0 =
1
(2π)N
2π
∫
0
dNφ0 Oi = 〈Oi〉0. (4.83)
Therefore, as in the quantum case, the average of a single deformation function turns
out to be time-independent.
Equivalence between the two approaches 43
Now, we consider the two-time average and expand it as follows:
〈Oi(t1)Oj(t2)〉0 =
1
(2π)N
2π
∫
0
dNφ0
∑
n
β(i)
n e
in·(φ0+ωt1)
∑
n′
β
(j)
n′ e
in′·(φ0+ωt2)
=
∑
n,n′
β(i)
n β
(j)
n′ e
in·ωt1ein
′·ωt2


1
(2π)N
2π
∫
0
dNφ0 e
i(n+n′)·φ0


=
∑
n,n′
β(i)
n β
(j)
n′ e
in·ωt1ein
′·ωt2δn′,−n
= β
(i)
0 β
(j)
0 +
∑
n 6=0
β(i)
n β
(j)
−ne
in·ω(t1−t2). (4.84)
Thus, equations (4.83) and (4.84) lead to
〈Oi(t1)Oj(t2)〉0 − 〈Oi(t)〉0〈Oj(t)〉0 =
∑
n 6=0
β(i)
n β
(j)
−ne
in·ω(t1−t2). (4.85)
Substituting (4.85) in (4.75), it is now easy to isolate the time dependence, just as in
the quantum case. Hence,
gij = −
∑
n 6=0
β(i)
n β
(j)
−n
0
∫
−∞
dt1
∞
∫
0
dt2 e
in·ω(t1−t2), (4.86)
and upon using the regularization (2.63)
0
∫
−∞
dt1
∞
∫
0
dt2 e
in·ω(t1−t2) = − 1
(n · ω)2 , (4.87)
we find that
gij =
∑
n 6=0
β
(i)
n β
(j)
−n
(n · ω)2 . (4.88)
This expression tells us that from the Fourier coefficients of the time-independent defor-
mation functions Oi, and the frequencies ωa, we can compute the classical metric. To
better understand the result, let us use (4.81) and rewrite it as
gij =
∑
n 6=0
〈Oie
−in·φ0〉0〈Oje
in·φ0〉0
(n · ω)2 . (4.89)
Remarkably, this expression for the classical metric is similar to equation (2.64) that
corresponds to the quantum metric. Now, we consider the following relation
H(q0(φ0, I; x), p0(φ0, I; x); x) = H(I; x), (4.90)
44 Classical analog of the quantum geometric tensor
and differentiate it with respect to xi. We get
N
∑
a=1
[
(
∂H
∂qa0
)
p0,x
(
∂qa0
∂xi
)
φ0,I
+
(
∂H
∂p0a
)
q0,x
(
∂p0a
∂xi
)
φ0,I
]
+
(
∂H
∂xi
)
q0,p0
=
(
∂H
∂xi
)
I
. (4.91)
The generators of parameter displacements Gi are readily identified through equations
(4.15) and (4.16). Thus,
N
∑
a=1
[
(
∂H
∂qa0
)
p0,x
(
∂Gi
∂p0a
)
q0,x
−
(
∂H
∂p0a
)
q0,x
(
∂Gi
∂qa0
)
p0,x
]
+Oi =
(
∂H
∂xi
)
I
. (4.92)
The first term is the Poisson bracket of H with the generator Gi, hence,
Oi =
(
∂H
∂xi
)
I
− {H,Gi}(q0,p0). (4.93)
Next, we take advantage of the canonical invariance of Poisson brackets and compute
them in terms of action-angle variables to get
Oi =
(
∂H
∂xi
)
I
+
N
∑
a=1
ωa
(
∂Gi
∂φa0
)
I,x
, (4.94)
where we have used the fact that ωa = ∂H/∂I. With this at hand, we can rewrite one
of the averages that appear in (4.89) as
〈Oie
−in·φ0〉0 =
(
∂H
∂xi
)
I
〈e−in·φ0〉0 +
N
∑
a=1
ωa
〈
∂Gi
∂φa0
e−in·φ0
〉
0
=
(
∂H
∂xi
)
I
δn,0 +
N
∑
a=1
ωa
〈
∂Gi
∂φa0
e−in·φ0
〉
0
. (4.95)
Now, we separate the ath degree of freedom in the average of the second term and
perform an integration by parts as follows:
〈
∂Gi
∂φa0
e−in·φ0
〉
0
=
1
(2π)N
2π
∫
0
dN−1φ0 e
−i
∑
b 6=a nbφ
b
0
2π
∫
0
dφa0
∂Gi
∂φa0
e−inaφa0
=
1
(2π)N
2π
∫
0
dN−1φ0 e
−i
∑
b 6=a nbφ
b
0

Gie
−inaφa0
∣
∣
∣
∣
2π
0
+ ina
2π
∫
0
dφa0Gie
−inaφa0


=
ina
(2π)N
2π
∫
0
dNφ0Gie
−in·φ0
= ina〈Gie
−in·φ0〉0, (4.96)
Equivalence between the two approaches 45
where in the third line we have used the fact that Gi is periodic in every φa0. Therefore,
substituting this into (4.95), we find that
〈Oie
−in·φ0〉0 =
(
∂H
∂xi
)
I
δn,0 +
(
N
∑
a=1
inaω
a
)
〈Gie
−in·φ0〉0
=
(
∂H
∂xi
)
I
δn,0 + i(n · ω)〈Gie
−in·φ0〉0. (4.97)
Similarly, the other average appearing in (4.89) turns out to be
〈Oje
in·φ0〉0 =
(
∂H
∂xj
)
I
δn,0 − i(n · ω)〈Gje
in·φ0〉0. (4.98)
Substituting these averages in (4.89) and noticing that n 6= 0, we arrive at
gij =
∑
n 6=0
〈Gie
−in·φ0〉0〈Gje
in·φ0〉0, (4.99)
so adding and subtracting the term with n = 0, we get
gij =
∑
n
〈Gie
−in·φ0〉0〈Gje
in·φ0〉0 − 〈Gi〉0〈Gj〉0. (4.100)
Expanding the first term, we find that
∑
n
〈Gie
−in·φ0〉0〈Gje
in·φ0〉0 =
∑
n
1
(2π)N
2π
∫
0
dNφ0Gie
−in·φ0 1
(2π)N
2π
∫
0
dNφ′0G
′
je
in·φ′0
=
1
(2π)N
1
(2π)N
2π
∫
0
dNφ0
2π
∫
0
dNφ′0GiG
′
j
∑
n
ein·(φ
′
0−φ0)
=
1
(2π)N
1
(2π)N
2π
∫
0
dNφ0
2π
∫
0
dNφ′0GiG
′
j(2π)
Nδ(N)(φ′0 − φ0)
=
1
(2π)N
2π
∫
0
dNφ0GiGj
= 〈GiGj〉0. (4.101)
Putting this result in (4.100), we arrive at
gij = 〈GiGj〉0 − 〈Gi〉0〈Gj〉0. (4.102)
Finally, due to the gauge invariance of the metric, we can pick any angle to carry out
the torus average, not necessarily the initial angle φ0. Thus,
gij = 〈GiGj〉 − 〈Gi〉〈Gj〉, (4.103)
which is precisely equation (4.38).
46 Classical analog of the quantum geometric tensor
4.4.2 The Hannay curvature
Now, we consider the Hannay curvature. Naturally, due to (4.80) being expressed in
terms of action-angle variables, we compute the Poisson bracket as
{Oi(t1),Oj(t2)} =
N
∑
a=1
[
∂Oi(t1)
∂φa0
∂Oj(t2)
∂Ia
− ∂Oi(t1)
∂Ia
∂Oj(t2)
∂φa0
]
. (4.104)
The average of the first term is
〈 N
∑
a=1
∂Oi(t1)
∂φa0
∂Oj(t2)
∂Ia
〉
0
=
∑
n,n′
N
∑
a=1
β(i)
n e
in·ωt1ina
∂
∂Ia
(
β
(j)
n′ e
in′·ωt2
)
〈ei(n+n′)·φ0〉0
=
∑
n,n′
N
∑
a=1
β(i)
n e
in·ωt1ina
∂
∂Ia
(
β
(j)
n′ e
in′·ωt2
)
δn′,−n
=
∑
n
N
∑
a=1
β(i)
n e
in·ωt1ina
∂
∂Ia
(
β
(j)
−ne
−in·ωt2
)
, (4.105)
and the average of the second term is obtained with t1 ↔ t2 and i↔ j:
〈 N
∑
a=1
∂Oi(t1)
∂Ia
∂Oj(t2)
∂φa0
〉
0
=
∑
n
N
∑
a=1
β(j)
n ein·ωt2ina
∂
∂Ia
(
β
(i)
−ne
−in·ωt1
)
= −
∑
n
N
∑
a=1
β
(j)
−ne
−in·ωt2ina
∂
∂Ia
(
β(i)
n e
in·ωt1) , (4.106)
where in the last line we made change n→ −n. Now, subtracting (4.106) from (4.105),
we recognize the derivative of a product, hence,
〈{Oi(t1),Oj(t2)}〉0 =
∑
n
N
∑
a=1
ina
∂
∂Ia
(
β(i)
n β
(j)
−ne
in·ω(t1−t2)
)
=
∑
n 6=0
N
∑
a=1
ina
∂
∂Ia
(
β(i)
n β
(j)
−ne
in·ω(t1−t2)
)
. (4.107)
The Hannay curvature then is
Fij =
∑
n 6=0
N
∑
a=1
ina
∂
∂Ia

β(i)
n β
(j)
−n
0
∫
−∞
dt1
∞
∫
0
dt2 e
in·ω(t1−t2)

 , (4.108)
and using the result of the time integral (4.87) we arrive at
Fij = −
∑
n 6=0
N
∑
a=1
ina
∂
∂Ia
[
β
(i)
n β
(j)
−n
(n · ω)2
]
, (4.109)
Equivalence between the two approaches 47
or
Fij = −
∑
n 6=0
N
∑
a=1
ina
∂
∂Ia
[〈Oie
−in·φ0〉0〈Oje
in·φ0〉0
(n · ω)2
]
. (4.110)
Taking into account equations (4.97) and (4.98), the Hannay curvature takes the form
Fij = −
∑
n 6=0
N
∑
a=1
ina
∂
∂Ia
(
〈Gie
−in·φ0〉0〈Gje
in·φ0〉0
)
. (4.111)
Considering again the null n = 0 term and expanding the product, we get
Fij = Aij +Bij, (4.112)
where
Aij = −
∑
n
N
∑
a=1
ina
〈
∂Gi
∂Ia
e−in·φ0
〉
0
〈Gje
in·φ0〉0 (4.113)
and
Bij = −
∑
n
N
∑
a=1
ina〈Gie
−in·φ0〉0
〈
∂Gj
∂Ia
ein·φ0
〉
0
. (4.114)
We manipulate the first term as follows:
Aij = −
∑
n
N
∑
a=1
ina
1
(2π)N
2π
∫
0
dNφ0
∂Gi
∂Ia
e−in·φ0
1
(2π)N
2π
∫
0
dNφ′0G
′
je
in·φ′0
=
∑
n
N
∑
a=1
1
(2π)N
2π
∫
0
dNφ0
[
∂Gi
∂Ia
∂
∂φa0
(e−in·φ0)
]
1
(2π)N
2π
∫
0
dNφ′0G
′
je
in·φ′0
=
N
∑
a=1
1
(2π)N
2π
∫
0
dNφ0


∂Gi
∂Ia
∂
∂φa0


1
(2π)N
2π
∫
0
dNφ′0G
′
j
(
∑
n
ein·(φ
′
0−φ0)
)




=
N
∑
a=1
1
(2π)N
2π
∫
0
dNφ0


∂Gi
∂Ia
∂
∂φa0


1
(2π)N
2π
∫
0
dNφ′0G
′
j(2π)
Nδ(N)(φ
′
0 − φ0)




=
N
∑
a=1
1
(2π)N
2π
∫
0
dNφ0
∂Gi
∂Ia
∂Gj
∂φa0
=
〈 N
∑
a=1
∂Gi
∂Ia
∂Gj
∂φa0
〉
0
. (4.115)
48 Classical analog of the quantum geometric tensor
Similarly, the second term yields
Bij = −
〈 N
∑
a=1
∂Gi
∂φa0
∂Gj
∂Ia
〉
0
. (4.116)
Thus, the Hannay curvature turns out to be
Fij = Aij +Bij =
〈 N
∑
a=1
(
∂Gi
∂Ia
∂Gj
∂φa0
− ∂Gi
∂φa0
∂Gj
∂Ia
)〉
0
, (4.117)
which reduces to
Fij = −〈{Gi, Gj}〉0. (4.118)
Once again, due to gauge invariance, we have
Fij = −〈{Gi, Gj}〉, (4.119)
which is equation (4.22).
Summarizing, the different ways of computing the classical metric and the Hannay
curvature are
1. Generator approach
gij(I; x) = 〈GiGj〉 − 〈Gi〉〈Gj〉, (4.120)
Fij(I; x) = −〈{Gi, Gj}〉. (4.121)
2. Decomposition in Fourier modes
gij(I; x) =
∑
n 6=0
〈Oie
−in·φ0〉0〈Oje
in·φ0〉0
(n · ω)2 , (4.122)
Fij(I; x) = −
∑
n 6=0
N
∑
a=1
ina
∂
∂Ia
[〈Oie
−in·φ0〉0〈Oje
in·φ0〉0
(n · ω)2
]
. (4.123)
3. Time-dependent approach
gij(I; x) = −
0
∫
−∞
dt1
∞
∫
0
dt2
(
〈Oi(t1)Oj(t2)〉0 − 〈Oi(t1)〉0〈Oj(t2)〉0
)
, (4.124)
Fij(I; x) =
0
∫
−∞
dt1
∞
∫
0
dt2 〈{Oi(t1),Oj(t2)}〉0. (4.125)
It is worth noticing the remarkable similarity between these expressions and those on
the quantum side (2.68 - 2.70).
Chapter 5
Examples of the generator approach to
the classical metric
In this chapter, we illustrate the application of the formulas to compute the quantum
connection, curvature, and metric as well as their classical counterparts. The classical
results are obtained under the generator approach and every system that is considered
presents a distinctive feature. We see how the classical metric contains all, or almost all
the information of the parameter space provided that we establish some identification
rules for the action variables. Furthermore, in some cases, the Berry curvature is zero,
and as a consequence, information about the parameter space cannot be read from this
quantity. Therefore, in these cases the quantum metric tensor and its classical analog
come into play and help one see the underlying parameter geometry. The results of this
chapter have been published in [67].
5.1 Generalized harmonic oscillator
The generalized harmonic oscillator is the archetypal system where one can exactly
calculate the connection, curvature, and metric. It was analyzed by Berry and Hannay
in their seminal works [8, 9]. In the first subsection, we present the standard results of
the generalized harmonic oscillator. In the second subsection, we add a linear term in the
position and show that the system is still exactly solvable regardless of the magnitude
of the coupling parameter.
5.1.1 Standard generalized harmonic oscillator
The Hamiltonian of this system is
Ĥ =
1
2
[
Xq̂2 + Y (q̂p̂+ p̂q̂) + Zp̂2
]
. (5.1)
The parameters are x = {xi} = (X, Y, Z) with i = 1, 2, 3, and in order to have periodic
motion we restrict ourselves to the subspace defined by XZ−Y 2 > 0. The wave function
49
50 Examples of the generator approach to the classical metric
is found to be
ψn(q; x) =
( ω
Z~
)1/4
χn
(
q
√
ω
Z~
)
exp
(
− iY q
2
2Z~
)
, (5.2)
where ω =
√
XZ − Y 2 and
χn(ξ) =
(
2nn!
√
π
)−1/2
e−ξ
2/2Hn(ξ) (5.3)
with Hn(ξ) the nth Hermite polynomial. The energy eigenvalues are En = (n+ 1/2)~ω
(n = 0, 1, 2, ...).
Now, the quantum connection, curvature, and metric can be computed using the
following property of the Hermite functions:
∞
∫
−∞
dξ ξ2χ2
n(ξ) = n+
1
2
. (5.4)
The components of the Berry connection (2.5) are
A
(n)
1 = 0, A
(n)
2 =
(
n+
1
2
)
1
2ω
, A
(n)
3 = −
(
n+
1
2
)
Y
2Zω
. (5.5)
From these, we get the components of the Berry curvature (2.9),
F
(n)
23 = −
(
n+
1
2
)
X
4ω3
, F
(n)
31 = −
(
n+
1
2
)
Y
4ω3
, F
(n)
12 = −
(
n+
1
2
)
Z
4ω3
. (5.6)
And, finally, using (5.2) we get the components of the quantum metric tensor,
g
(n)
ij =
n2 + n+ 1
32ω4


Z2 −2Y Z 2Y 2 −XZ
−2Y Z 4XZ −2XY
2Y 2 −XZ −2XY X2

 . (5.7)
We see that these results agree with those found for the same system in Chapter 3 using
the Wigner function formalism.
On the other hand, for the classical calculation we need first to express the solution
of the equations of motion in terms of action-angle variables. We get [69]
q(φ, I; x) =
√
2IZ
ω
sinφ, p(φ, I; x) =
√
2IZ
ω
(
−Y
Z
sinφ+
ω
Z
cosφ
)
, (5.8)
so that in terms of these variables the Hamiltonian has the form H = Iω. In order to
find the generators Gi, we can integrate their defining equations (4.15,4.16) which yields
G1 =− Y
4ω2
q2 − Z
4ω2
qp,
G2 =
X
2ω2
q2 +
Y
2ω2
qp,
G3 =− XY
4Zω2
q2 +
XZ − 2Y 2
4Zω2
qp. (5.9)
Generalized harmonic oscillator 51
Of course, we could have found the generating function S(q, I, x) of the canonical trans-
formation (q, p) → (φ, I), and then use Gi = (p∂iq)φ,I − ∂iSφ,I . The generating function
in terms of action-angle variables is [69]
S(φ, I; x) = −IY
ω
sin2 φ+ I(φ+ sinφ cosφ). (5.10)
Thus, the generators are
G1 =− IZ
2ω2
sinφ cosφ,
G2 =
I sinφ
ω2
(Y cosφ+ ω sinφ) ,
G3 =
I sinφ
2Zω2
[
(XZ − 2Y 2) cosφ− 2Y ω sinφ
]
. (5.11)
Upon substitution of q(φ, I; x) and p(φ, I; x) into (5.9) we reproduce the above expres-
sions. The components of the classical Hannay connection Ai = 〈Gi〉 are
A1 = 0, A2 =
I
2ω
, , A3 = − IY
2Zω
. (5.12)
Thus, the curvature turns out to be
F23 = − IX
4ω3
, F31 = − IY
4ω3
, F12 = − IZ
4ω3
, (5.13)
and the metric tensor has the form
gij =
I2
32ω4


Z2 −2Y Z 2Y 2 −XZ
−2Y Z 4XZ −2XY
2Y 2 −XZ −2XY X2

 . (5.14)
Some observations are in order:
1. The quantum results might have been obtained using the expressions for the gen-
erators in terms of q and p and quantizing with the appropriate symmetrization
taking care of the Hermitian nature of the generators Ĝi.
2. We could have also found the curvature directly by taking the Poisson brackets of
the generators in the form of (5.9) or (5.11) since they are canonical invariants.
3. The action variable is quantized as I = (n+1/2)~. The substitution of this result
in the classical Hannay connection and curvature yields
A
(n)
i =
Ai
~
, F
(n)
ij =
Fij
~
, (5.15)
which are precisely the relations found by Berry in [9]. See also (4.76).
52 Examples of the generator approach to the classical metric
4. For the metrics, we have to use the identification I2 → (n2 + n+ 1)~2 to get
g
(0)
ij =
gij
~2
, (5.16)
which is the relation (4.74). The quantization rule I = (n + 1/2)~ does not give
the correct result for the metric tensor since I2 = (n2 + n + 1/4)~2. We will see
that this mismatch is a common feature to all the examples that we consider,
indicating that powers of the action variable should be quantized by adding some
correction. It is important to remark that even in the case of the harmonic oscillator
H = (p2/2m+ kq2/2) this effect is observed.
5. The determinants of both the quantum and classical metric are zero, which indi-
cates that the Hamiltonian involves more parameters than the effective ones. This
can be seen through a change of parameters x → x′ that has the same form as
that of the Cartesian to spherical coordinates. The effect of this transformation in
parameter space is to make zero one row and column of the metric tensors. In fact,
the metrics have rank two and hence, in order to have non-vanishing determinants,
we must leave one of the parameters fixed (but different from zero).
5.1.2 Linear term
Now, we add a linear term in the position with coupling parameter W to the previous
Hamiltonian:
Ĥ =
1
2
[
Xq̂2 + Y (q̂p̂+ p̂q̂) + Zp̂2
]
+Wq̂. (5.17)
If we set W = 0, we recover the generalized harmonic oscillator. We take as our parame-
ters x = {xi} = (W,X, Y, Z) with i = 0, 1, 2, 3. Now, the Schrödinger equation acquires
an extra term
− Z~2
2
d2ψn
dq2
− i~Y q
dψn
dq
+
(
Xq2
2
− i~
Y
2
)
ψn +Wqψn = Enψn, (5.18)
and this modifies the wave function
ψn(q; x) =
( ω
Z~
)1/4
χn
[(
q +
WZ
ω2
)
√
ω
Z~
]
exp
(
− iY q
2
2Z~
)
. (5.19)
Through the use of the properties of Hermite polynomials we calculate the connection,
curvature, and metric. The connection is
A
(n)
0 = A
(n)
1 = 0, A
(n)
2 =
n+ 1
2
2ω
+
W 2Z
2~ω4
, A
(n)
3 = −
(
n+ 1
2
)
Y
2Zω
− W 2Y
2~ω4
. (5.20)
Generalized harmonic oscillator 53
From here, we get the six independent components of the curvature which are displayed
in matrix form as
F
(n)
ij =
n+ 1
2
4ω3




0 0 0 0
0 0 −Z Y
0 Z 0 −X
0 −Y X 0




+
1
~ω6




0 0 WZω2 −WY ω2
0 0 −W 2Z2 W 2Y Z
−WZω2 W 2Z2 0 −W 2Y 2
WY ω2 −W 2Y Z W 2Y 2 0




.
(5.21)
The metric tensor turns out to be
g
(n)
ij =
n2 + n+ 1
32ω4




0 0 0 0
0 Z2 −2Y Z 2Y 2 −XZ
0 −2Y Z 4XZ −2XY
0 2Y 2 −XZ −2XY X2




+
n+ 1
2
~ω7




Zω4 −WZ2ω2 2WY Zω2 −WY 2ω2
−WZ2ω2 W 2Z3 −2W 2Y Z2 W 2Y 2Z
2WY Zω2 −2W 2Y Z2 W 2Z(3Y 2 +XZ) −W 2Y (Y 2 +XZ)
−WY 2ω2 W 2Y 2Z −W 2Y (Y 2 +XZ) W 2XY 2




. (5.22)
We see that taking W = 0 in the above expressions and eliminating the corresponding
row and column of the tensors, the results match the standard generalized harmonic
oscillator.
For the classical analogs, we use the position and momentum in terms of action-angle
variables which can be found almost in the same way as in the previous example.
q(φ, I; x) =
√
2IZ
ω
sinφ− WZ
ω2
, p(φ, I; x) =
√
2IZ
ω
(
−Y
Z
sinφ+
ω
Z
cosφ
)
+
WY
ω2
,
(5.23)
so that the Hamiltonian now reads H = Iω −W 2Z/2ω2, with ω =
√
XZ − Y 2. The
generating function S of the transformation to action-angle variables is
S(φ, I; x) = − Y
2Z
(
√
2IZ
ω
sinφ− WZ
ω2
)2
+ I(φ+ sinφ cosφ), (5.24)
and the generators Gi written in compact form are
Gi(φ, I; x) = fi(x)qp+ gi(x)q
2 + hi(x)p+
Y
Z
hi(x)q, (5.25)
where q = q(φ, I; x) and p = p(φ, I; x) are given by (5.23) and
fi(x) =
ω
2Z
∂
∂xi
(
Z
ω
)
,
gi(x) =
Y
Z
fi(x) +
1
2
∂
∂xi
(
Y
Z
)
,
hi(x) =
W
2ω
∂
∂xi
(
Z
ω
)
− ∂
∂xi
(
WZ
ω2
)
. (5.26)
54 Examples of the generator approach to the classical metric
It can readily be verified that if the parameter W is fixed to zero, we recover the gener-
ators (5.9). With these expressions at hand, we can get the connection,
A0 = A1 = 0, A2 =
I
2ω
+
W 2Z
2ω4
, A3 = − IY
2Zω
− W 2Y
2ω4
, (5.27)
the curvature,
Fij =
I
4ω3




0 0 0 0
0 0 −Z Y
0 Z 0 −X
0 −Y X 0




+
1
ω6




0 0 WZω2 −WY ω2
0 0 −W 2Z2 W 2Y Z
−WZω2 W 2Z2 0 −W 2Y 2
WY ω2 −W 2Y Z W 2Y 2 0




,
(5.28)
and the metric tensor,
gij =
I2
32ω4




0 0 0 0
0 Z2 −2Y Z 2Y 2 −XZ
0 −2Y Z 4XZ −2XY
0 2Y 2 −XZ −2XY X2




+
I
ω7




Zω4 −WZ2ω2 2WY Zω2 −WY 2ω2
−WZ2ω2 W 2Z3 −2W 2Y Z2 W 2Y 2Z
2WY Zω2 −2W 2Y Z2 W 2Z(3Y 2 +XZ) −W 2Y (Y 2 +XZ)
−WY 2ω2 W 2Y 2Z −W 2Y (Y 2 +XZ) W 2XY 2




. (5.29)
Notice again the different quantization of powers of the action variable. In this case,
the action variable in the metric tensor appears both linearly and squared, but only the
squared action is quantized in a distinct manner. Finally, it is easy to find that the
determinants of the metrics (5.22) and (5.29) are zero and their rank is three.
5.2 Quartic anharmonic oscillator
Here, we consider the Hamiltonian of the quartic anharmonic oscillator
Ĥ =
p̂2
2m
+
k
2
q̂2 +
λ
4!
q̂4 (5.30)
with parameters x = {xi} = (m, k, λ), i = 1, 2, 3. We focus on a perturbative treatment
for 0 < λ ≪ 1, since the analytical solutions for the eigenvalue problem cannot be
obtained. Also, we restrict ourselves to considering only the ground state wave function
and its energy in powers of λ up to third order.
First of all, notice that the definition of the Berry connection (2.12) and curvature
(2.13) requires that we take the imaginary part of an inner product [4]. However, since
the wave function that we consider is real, we have as a result a zero connection, and
thus, a zero curvature. This will happen too in the remaining examples.
Quartic anharmonic oscillator 55
The ground state energy and wave function of Hamiltonian (5.30) can be obtained
using the standard perturbation theory [77, 78]. Nevertheless, through the more efficient
non-linearization method [79] we can easily find corrections to arbitrarily large powers
of λ. The dimensionless form of the eigenvalue problem considered in [79] is
(
− d2
dX2
+X2 + γX4
)
ψ(X) = ǫψ(X). (5.31)
Of course, we can return to our eigenvalue problem with all the dimensions through the
following change of variables
γ =
~λ
12m2ω3
0
, X =
√
mω0
~
q, ǫ =
2E
~ω0
, (5.32)
where ω0 =
√
k/m. The non-normalized ground state wave function for the dimension-
less Hamiltonian is, to third order in γ,
ψ(X) = e−
X2
2
[
1− 1
8
γX2
(
X2 + 3
)
+
1
384
γ2X2
(
3X6 + 26X4 + 93X2 + 252
)
− 1
3072
γ3X2
(
X10 + 17X8 + 141X6 + 813X4 + 2916X2 + 7992
)
]
+O(γ4),
(5.33)
and the corresponding energy is
ǫ = 1 +
3
4
γ − 21
16
γ2 +
333
64
γ3 +O(γ4). (5.34)
For more details about the features of the asymptotic series, we refer the reader to the
standard references of Bender and Wu [80, 81]. After returning to our original variables
and normalizing, we find that
ψ(q; x) = e−
mω0
2~
q2
[
4
√
mω0
π~
− λ
P1(q; x)
384 4
√
πm7ω11
0 ~5
+ λ2
P2(q; x)
884736 4
√
πm15ω23
0 ~9
− λ3
P3(q; x)
339738624 4
√
πm23ω35
0 ~13
]
+O(λ4), (5.35)
E0 =
~ω0
2
+ λ
~
2
32m2ω2
0
− λ2
7~3
1536m4ω5
0
+ λ3
37~4
24576m6ω8
0
+O(λ4). (5.36)
where
P1(q; x) =4m2ω2
0q
4 + 12~mω0q
2 − 9~2,
P2(q; x) =48m4ω4
0q
8 + 416~m3ω3
0q
6 + 1272~2m2ω2
0q
4 + 3384~3mω0q
2 − 4677~4,
P3(q; x) = 64m6ω6
0q
12 + 1088~m5ω5
0q
10 + 8592~2m4ω4
0q
8
+ 48288~3m3ω3
0q
6 + 154524~4m2ω2
0q
4 + 419076~5mω0q
2 − 729153~6. (5.37)
56 Examples of the generator approach to the classical metric
To compute the quantum metric tensor, it is sufficient to consider the expression
g
(0)
ij = 〈∂iψ|∂jψ〉 − 〈∂iψ|ψ〉〈ψ|∂jψ〉 (5.38)
due to the reality of the wave function. We find the components
g
(0)
11 =
1
32m2
− λ
3~
512
√
m5k3
+ λ2
59~2
16384m3k3
+O(λ3),
g
(0)
12 =
1
32mk
− λ
7~
512
√
m3k5
+ λ2
143~2
16384m2k4
+O(λ3),
g
(0)
13 =
~
128
√
m3k3
− λ
21~2
4096m2k3
+ λ2
2353~3
589824
√
m5k9
+O(λ3),
g
(0)
22 =
1
32k2
− λ
11~
512
√
mk7
+ λ2
785~2
49152mk5
+O(λ3),
g
(0)
23 =
~
128
√
mk5
− λ
89~2
12288mk4
+ λ2
3841~3
589824
√
m3k11
+O(λ3),
g
(0)
33 =
13~2
6144mk3
− λ
31~3
12288
√
m3k9
+ λ2
57227~4
21233664m2k6
+O(λ3). (5.39)
It is important to remark that the metric tensor is correct up to one order less than the
wave function since there are components which require the calculation of a derivative
with respect to λ. Another observation can be made regarding the determinant of the
metric. We can easily find that
det g
(0)
ij = O(λ3). (5.40)
However, if we fix the mass and only take as our parameters x = {xi} = (k, λ) with
i = 1, 2, the determinant of this reduced 2× 2 metric is
det g
(0)red
ij =
~
2
196608mk5
− λ
35~3
3145728
√
m3k13
+ λ2
48935~4
2717908992m2k8
+O(λ3). (5.41)
Let us now turn to the classical calculation. The tool that we use is canonical
perturbation theory [76, 82]. The central idea of this method is that we can divide our
Hamiltonian in two parts:
H = H0 + λH1, (5.42)
where 0 < λ ≪ 1. It is assumed that we can find the position and momentum for the
system with unperturbed Hamiltonian H0 in terms of action-angle variables (φ0, I0). In
this case, we take
H0 =
p2
2m
+
k
2
q2, (5.43)
and
H1 =
q4
4!
. (5.44)
Quartic anharmonic oscillator 57
The solution (q, p) in terms of old action-angle variables (φ0, I0) is well-known:
q(φ0, I0; x) =
√
2I0
mω0
sinφ0, p(φ0, I0; x) =
√
2mω0I0 cosφ0, (5.45)
where ω0 =
√
k/m is the unperturbed frequency. The problem is to find the canonical
transformation from old action-angle variables (φ0, I0) to the new set (φ, I) such that
now the perturbed Hamiltonian H can be written only in terms of I, i.e.,
H(q(φ, I; x), p(φ, I; x); x) = E(I; x). (5.46)
This canonical transformation is generated by the type 2 function W (φ0, I; x),
W (φ0, I; x) = φ0I + λW1(φ0, I; x) + λ2W2(φ0, I; x) + λ3W3(φ0, I; x) +O(λ4), (5.47)
and the Hamiltonian E(I; x) will be expressed as
E(I; x) = E0(I; x) + λE1(I; x) + λ2E2(I; x) + λ3E3(I; x) +O(λ4). (5.48)
From the equations of the canonical transformation, φ = ∂W/∂I and I0 = ∂W/∂φ0, we
can find that
φ(φ0, I; x) = φ0 + λ
∂W1(φ0, I; x)
∂I
+ λ2
∂W2(φ0, I; x)
∂I
+ λ3
∂W3(φ0, I; x)
∂I
+O(λ4) (5.49)
and
I0(φ0, I; x) = I + λ
∂W1(φ0, I; x)
∂φ0
+ λ2
∂W2(φ0, I; x)
∂φ0
+ λ3
∂W3(φ0, I; x)
∂φ0
+O(λ4). (5.50)
The results of canonical perturbation theory are
Eµ(I; x) = 〈Φµ(φ0, I; x)〉0, (5.51)
ω0(I; x)
∂Wµ(φ0, I; x)
∂φ0
= 〈Φµ(φ0, I; x)〉0 − Φµ(φ0, I; x), µ = 1, 2, ... , (5.52)
where ω0(I; x) = (∂H0/∂I0)I0=I and the symbol 〈·〉0 denotes the average with respect
to the angle variable φ0. In our case, ω0(I; x) = ω0 =
√
k/m. The function Φ0 is
just the unperturbed Hamiltonian H0 evaluated at the new action variable I, which is
Φ0(I; x) = Iω0. The function Φ1 is the perturbation H1 written in terms of the old angle
φ0 and the new action I, i.e.,
Φ1(φ0, I; x) =
q4(φ0, I; x)
4!
=
I2 sin4 φ0
6m2ω2
0
. (5.53)
The functions Φ2 and Φ3 can be obtained by the Taylor series expansion of the Hamil-
tonian, and they are
Φ2(φ0, I; x) =
∂W1
∂φ0
∂H1
∂I
, (5.54)
58 Examples of the generator approach to the classical metric
Φ3(φ0, I; x) =
1
2
(
∂W1
∂φ0
)2
∂2H1
∂I2
+
∂W2
∂φ0
∂H1
∂I
. (5.55)
Also, since we have only one degree of freedom, the Wµ can be easily calculated by
integrating (5.52). It turns out that
Φ2(φ0, I; x) = −I
3 sin4 φ0
(
8 sin4 φ0 − 3
)
144m4ω5
0
,
Φ3(φ0, I; x) =
I4 sin4 φ0
(
320 sin8 φ0 − 144 sin4 φ0 − 25
)
13824m6ω8
0
, (5.56)
and
W1(φ0, I; x) =
I2(8 sin 2φ0 − sin 4φ0)
192m2ω3
0
,
W2(φ0, I; x) =
I3(−384 sin 2φ0 + 132 sin 4φ0 − 32 sin 6φ0 + 3 sin 8φ0)
55296m4ω6
0
,
W3(φ0, I;x) =
I4(9264 sin 2φ0 − 4101 sin 4φ0 + 1624 sin 6φ0 − 441 sin 8φ0 + 72 sin 10φ0 − 5 sin 12φ0)
5308416m6ω9
0
.
(5.57)
With this at hand, we are able to calculate the energy series:
E = Iω0 + λ
I2
16m2ω2
0
− λ2
17I3
2304m4ω5
0
+ λ3
125I4
73728m6ω8
0
+O(λ4). (5.58)
Notice that all the expressions we have found so far are written in terms of the old angle
variable φ0 and the new action variable I. In order to compute the metric tensor, we
need to find the generators Gi = p(∂iq)φ,I − (∂iS)φ,I , where the derivatives are taken at
fixed new action-angle variables (φ, I). This requires the manipulation of the derivatives
to write them in the most convenient form for computation. Another issue to address is
that we must find the generating function S of the transformation from variables (q, p)
to (φ, I) and it is not evident how to do it, since we only have the function W that
performs the transformation (φ0, I0) → (φ, I).
The first issue is solved by noting that given a function f(φ0, I; x), we can differentiate
the expression f(φ0, I; x) = f(φ(φ0, I; x), I; x) with respect to the parameters x to find
(
∂f
∂xi
)
φ0,I
=
(
∂f
∂φ
)
I,x
(
∂φ
∂xi
)
φ0,I
+
(
∂f
∂xi
)
φ,I
. (5.59)
Now, we still have to apply a cyclic property to write the derivative of f with respect to
φ at our convenience. This can be done through the identity
(
∂f
∂φ
)
I,x
=
(∂f/∂φ0)I,x
(∂φ/∂φ0)I,x
. (5.60)
Quartic anharmonic oscillator 59
Hence, we are finally able to express everything in terms of derivatives that we can
calculate:
(
∂f
∂xi
)
φ,I
=
(
∂f
∂xi
)
φ0,I
− (∂f/∂φ0)I,x
(∂φ/∂φ0)I,x
(
∂φ
∂xi
)
φ0,I
. (5.61)
The second issue can be solved by noting that we already have the position and mo-
mentum in terms of the old action-angle variables (5.45). This leads us into thinking that
we may first attempt to perform a transformation (q, p) → (φ0, I0) through a generating
function S0(q, I0; x), and then perform the transformation (φ0, I0) → (φ, I) through our
W (φ0, I; x). The resulting generating function S for the composition of two canonical
transformations is [71, 72]
S(q, I; x) = S0(q, I0; x) +W (φ0, I; x)− φ0I0, (5.62)
and the function S0 (which we write in terms of old action-angle variables for convenience)
is
S0(φ0, I0; x) = I0(φ0 + sinφ0 cosφ0). (5.63)
With these results at hand, we can now compute the classical metric gij = 〈GiGj〉 −
〈Gi〉〈Gj〉 through the use of (5.61) making f = q or f = S. The generators Gi are given
in Appendix A. Notice that since we have everything written in terms of the old angle
φ0, we need to calculate the averages that appear in the metric by making the change of
variable
φ = φ(φ0, I; x), dφ =
(
∂φ
∂φ0
)
I,x
dφ0. (5.64)
After taking all this into account, we find the components
g11 =
I2
32m2
− λ
I3
256
√
m5k3
+ λ2
47I4
32768m3k3
+O(λ3),
g12 =
I2
32mk
− λ
7I3
768
√
m3k5
+ λ2
347I4
98304m2k4
+O(λ3),
g13 =
I3
192
√
m3k3
− λ
103I4
49152m2k3
+ λ2
15I5
16384
√
m5k9
+O(λ3),
g22 =
I2
32k2
− λ
11I3
768
√
mk7
+ λ2
1919I4
294912mk5
+O(λ3),
g23 =
I3
192
√
mk5
− λ
439I4
147456mk4
+ λ2
7I5
4608
√
m3k11
+O(λ3),
g33 =
65I4
73728mk3
− λ
89I5
147456
√
m3k9
+ λ2
130621I6
382205952m2k6
+O(λ3). (5.65)
We can easily see that in order to match the quantum and classical metrics, we should
set for the ground state: I2/~2 = 1, I3/~3 = 3/2. However, for powers greater or equal
60 Examples of the generator approach to the classical metric
to 4, we cannot find the appropriate quantization. For instance, equating each of the
terms of both metrics that contain I4, we find six different results:
I4
~4
≈ 2.4, 2.4328, 2.4466, 2.4544, 2.4726, 2.5106, (5.66)
whereas for I5, we have
I5
~5
≈ 4.1797, 4.2868, 4.3574. (5.67)
Moreover, this quantization does not allow us to recover the energy (5.36). Indeed, it
can be seen that the energy series follows the quantization
I
~
=
(
n+
1
2
)
,
I2
~2
=
(
n+
1
2
)2
+
1
4
,
I3
~3
=
(
n+
1
2
)3
+
67
68
(
n+
1
2
)
, (5.68)
which for the ground state reduces to
I
~
=
1
2
,
I2
~2
=
1
2
,
I3
~3
=
21
34
. (5.69)
This result was determined by proposing a polynomial in n of the same degree as the
power of the action variable and using the expression for the energies of the first four
excited states given in [83]. It is important to comment that we have found the non-
trivial result that through classical calculations we have reproduced the structure of
the quantum metric. The mismatch is present since the quantum metric includes the
quantum perturbative effects which the classical perturbative method fails to reproduce,
hence the need of the empirical rules of quantization of the action variable. Finally, we
should remark that the method based on canonical perturbation theory applied to the
generalized harmonic oscillator with a linear term provides the same results.
5.3 Two-dimensional isotropic harmonic oscillator in
polar coordinates
This is a very illustrative example since the isotropic harmonic oscillator in two dimen-
sions can be effectively reduced to a one degree of freedom by using the conservation of
angular momentum. For this system, the Schrödinger equation reads
− ~
2
2m
(
∂2
∂r2
+
1
r
∂
∂r
+
1
r2
∂2
∂θ2
)
Ψ+
k
2
r2Ψ = EΨ. (5.70)
We take as our parameters the mass and the spring constant x = {xi} = (m, k) with
i = 1, 2. The wave function can be separated in an angular part Θ(θ) and a radial
Two-dimensional isotropic harmonic oscillator in polar coordinates 61
part ψ(r; x), such that Ψ(r, θ; x) = ψ(r; x)Θ(θ). In particular, for the ground state with
energy E = ~ω, the radial function is
ψ(r; x) =
√
2
√
mk
~
e−
√
mk
2~
r2 , (5.71)
whereas the angular part is only Θ(θ) = 1/
√
2π. By convention, the normalization is
taken separately so that the condition
〈Ψ|Ψ〉 =
2π
∫
0
dθ |Θ(θ)|2
∞
∫
0
dr r|ψ(r; x)|2 = 1 (5.72)
means that every integral should be equal to one. Thus, in the simple case of the ground
state the effective wave function is ψ(r; x) and the inner product of two functions is given
by
〈f |g〉 =
∞
∫
0
dr rf ∗(r; x)g(r; x). (5.73)
Since the wave function (5.71) is not complex, we automatically know that the connec-
tion, and consequently, the curvature, are zero. On the other hand, the metric tensor
takes the form
g
(0)
ij =
1
16
(
1
m2
1
mk
1
mk
1
k2
)
. (5.74)
The classical calculation is more involved, since it requires the knowledge of the equa-
tion of the orbit. We will follow an explicit approach which will be useful to summarize
some steps in the next example. The first step is to obtain the action-angle variables for
the system. The Hamiltonian is
H =
p2r
2m
+
p2θ
2mr2
+
k
2
r2. (5.75)
Using conservation of energy and angular momentum, we can see that
pr =
√
2mE −m2ω2r2 − L2/r2, pθ = L = constant, (5.76)
where ω =
√
k/m. From here, we get the angular action variable,
Iθ =
1
2π
∮
dθ pθ = |L|. (5.77)
On the other hand, the radial action variable is defined as
Ir =
1
2π
∮
dr pr =
1
2π
∮
dr
√
2mE −m2ω2r2 − L2/r2. (5.78)
62 Examples of the generator approach to the classical metric
A standard procedure to solve this integral (which will be used in the Coulomb problem
too) is to rewrite it as the sum of three integrals, M1, M2 and M3 [69]
Ir =
M1 +M2 +M3
2π
, (5.79)
where
M1 =
∮
dr
mE −m2ω2r2√
A
, M2 =
∮
dr
mE√
A
, M3 = −
∮
dr
L2
r2
√
A
, (5.80)
with A = 2mE−m2ω2r2−L2/r2. The first term is the integral of a total differential, so
M1 =
∮
d
(
r
√
2mE −m2ω2r2 − L2/r2
2
)
= 0. (5.81)
The second integral, M2, appears precisely when, through conservation of energy, one
tries to solve for the motion r = r(t), i.e.,
m
2
(
dr
dt
)2
+
mω2
2
r2 +
L2
2mr2
= E. (5.82)
This leads to
√
2
m
t =
∫
dr
√
E − mω2
2
r2 − L2
2mr2
(5.83)
We use the change of variable s = r2 to turn the integral into
√
m
2
E
2
∫
ds
√
Es− mω2
2
s2 − L2
2m
. (5.84)
Since this leaves a second degree polynomial inside the radical, we can make use of the
trigonometric substitution
s =
E
mω2
(
1−
√
1− ω2L2
E2
cosα
)
(5.85)
to simplify everything. Evaluating the indefinite integral in a cycle, this becomes
M2 =
E
2ω
∮
dα =
E
2ω
(2π) =
Eπ
ω
, (5.86)
since (5.85) indicates that in the phase space corresponding to r, the angle α moves 2π
until r returns to its initial value. Now, notice that M3 is the quadrature corresponding
to the orbit equation
L2
2mr4
(
dr
dθ
)2
+
mω2
2
r2 +
L2
2mr2
= E. (5.87)
Two-dimensional isotropic harmonic oscillator in polar coordinates 63
To solve this integral, we make the usual change of variable u = 1/r and then we set
u2 =
mE
L2
(
1 +
√
1− ω2L2
E2
cos β
)
. (5.88)
When we evaluate it in a cycle, we find
M3 = −1
2
∮
dβ L = −|L|
2
(2π) = −|L|π, (5.89)
because (5.88) returns to its initial value after β has advanced 2π. So, we conclude that
Ir =
1
2π
(
Eπ
ω
− |L|π
)
=
1
2
(
E
ω
− |L|
)
. (5.90)
By inverting this relation, we get the energy expressed in terms of the action variables
as
E = ω(2Ir + Iθ). (5.91)
The frequencies associated with the radial and angular motion are, respectively,
ωr =
∂E
∂Ir
= 2ω, ωθ =
∂E
∂Iθ
= ω. (5.92)
We would like to express the energy in terms of a single action variable which is the
one that will correspond to the principal quantum number [76]. In order to do this, we
consider the canonical transformation (φr, φθ, Ir, Iθ) → (φ1, φ2, I1, I2) generated by the
type 2 function
F2(φr, φθ, I1, I2) =
(
φθ −
φr
2
)
I1 + φrI2. (5.93)
The equations of the transformation imply that
E = 2ωI2, φ2 = φr, (5.94)
with ω1 = 0 and ω2 = 2ω. Therefore, we have reduced our system to one effective degree
of freedom. In what follows, we refer to I2 only as I, and to φ2 as φ.
Now we need to construct the generating function following a procedure that parallels
the previous one. After the elimination of the angular degree of freedom through the
conservation of angular momentum, we find that the function S which is the solution to
the Hamilton-Jacobi equation has the form
S(r, E; x) =
∫
dr
√
2mE −m2ω2r2 − L2/r2. (5.95)
We can read the result from the previous paragraphs:
S(r, E; x) =
r
√
2mE −m2ω2r2 − L2/r2
2
+
E
2ω
α(r, E; x)− |L|
2
β(r, E; x), (5.96)
64 Examples of the generator approach to the classical metric
where α and θ should be substituted from (5.85) and (5.88) in terms of r, E and the
parameters x. If we compare (5.85) with (5.83), we get that
α = 2ωt, (5.97)
and thus we find that the solution for the motion r(t) is given by
r2(t) =
E
mω2
(
1−
√
1− ω2L2
E2
cos 2ωt
)
. (5.98)
From here, it is seen that α = 2ωt is the angle variable φ = φr corresponding to the
radial degree of freedom. Furthermore, from (5.87) we find that β = 2θ, so the orbit
r(θ) is given by
1
r2(θ)
=
mE
L2
(
1 +
√
1− ω2L2
E2
cos 2θ
)
, (5.99)
which represents an ellipse with its geometrical center located at the origin. In order
to proceed with the calculation of the classical analog of the metric tensor, we need to
express the position r, the momentum pr, and the generating function S in terms of
action-angle variables. The rewriting of (5.98) using (5.94) results in
r(φ, I; x) =
√
2I√
mk
(
1−
√
1− L2/4I2 cosφ
)1/2
, (5.100)
and, since pr = mṙ,
pr(φ, I; x) =
√
2I
√
mk
√
1− L2/4I2 sinφ
(
1−
√
1− L2/4I2 cosφ
)1/2
. (5.101)
Moreover, the generating function becomes
S(φ, I; x) = I
(
1− L2/4I2
)
sinφ+ Iφ− |L|θ(φ, I; x). (5.102)
Now, we must relate the “physical angle” θ with the angle variable φ. A straightforward
comparison of the solution r(φ, I; x) and the orbit r(θ, I; x) shows that
1
1−
√
1− L2/4I2 cosφ
=
4I2
L2
(1 +
√
1− L2/4I2 cos 2θ). (5.103)
Hence, we see that the “physical angle” θ does not depend on the parameters x, and this
leads us to conclude that the generating function does not depend on x either, i.e.,
S(φ, I) = I
(
1− L2/4I2
)
sinφ+ Iφ− |L|θ(φ, I). (5.104)
Two-dimensional attractive Coulomb problem 65
Therefore, the generators Gi are reduced to Gi = pr∂ir. Remembering that x1 = m and
x2 = k, we find that
G1 = −
√
1− L2/4I2I sinφ
2m
= −rpr
4m
, (5.105)
G2 = −
√
1− L2/4I2I sinφ
2k
= −rpr
4k
. (5.106)
Finally, this allows us to compute the classical metric tensor which has the form
gij =
4I2 − L2
32
(
1
m2
1
mk
1
mk
1
k2
)
. (5.107)
According to the EBK quantization rule [74], we should have L = 0 and I = ~ for the
ground state, hence
gij(L = 0, I = ~) =
~
2
8
(
1
m2
1
mk
1
mk
1
k2
)
, (5.108)
and we see again that it differs from (5.74) by a factor of 2.
5.4 Two-dimensional attractive Coulomb problem
Now we address the two-dimensional attractive Coulomb problem whose Schrödinger
equation is
− ~
2
2m
(
∂2
∂r2
+
1
r
∂
∂r
+
1
r2
∂2
∂θ2
)
Ψ− k
r
Ψ = EΨ (5.109)
with k > 0. The parameters that we consider are x = {xi} = (m, k) with i = 1, 2.
According to Zaslow and Zandler [84], the spectrum is given by
En = − mk2
2~2
(
n− 1
2
)
2
, n = 1, 2, 3, ... . (5.110)
Just as in the previous case, for the ground state, separation of variables dictates that
the angular function Θ(θ) is 1/
√
2π. On the other hand, the radial wave function is
ψ(r; x) =
4mk
~2
e−
2mk
~2
r. (5.111)
We know that this will yield a zero connection so we move to the calculation of the
quantum metric tensor. The inner product for this Hilbert space has the same form as
(5.73), and this gives the metric tensor
g
(0)
ij =
1
2
(
1
m2
1
mk
1
mk
1
k2
)
. (5.112)
66 Examples of the generator approach to the classical metric
For the classical computation, we consider the Hamiltonian
H =
p2r
2m
+
L2
2mr2
− k
r
. (5.113)
We know from the previous section that in order to calculate the classical metric, we
require the motion as a function of time, r(t), and the orbit equation, r(θ). However, we
know that r(t) cannot be explicitly obtained for the Coulomb problem [76], so we must
work with the eccentric anomaly ψ rather than with the angle variable φ. The equation
r(t) is obtained, in principle, in two steps. First, we use Kepler’s equation
ωt = ψ − e sinψ, (5.114)
to find ψ(t). Here, ω = 1
k
√
−8E3
m
is the frequency of revolution (remember that E < 0
for the bounded problem), and e =
√
1 + 2EL2
mk2
is the eccentricity of the ellipse. Then,
we substitute this result in the equation that defines the eccentric anomaly
r = − k
2E
(1− e cosψ). (5.115)
It is straightforward to prove that the angle variable that corresponds to the radial degree
of freedom is φr = ωt, just as we would anticipate from (5.114). On the other hand, the
orbit equation r(θ) is given by
r(θ) =
L2
mk
1
1 + e cos θ
, (5.116)
and we know that it represents an ellipse with one of its foci located at the origin. We
now use the same technique as the previous section to find the action variables for the
system [69]. They are
Ir =
√
−mk
2
2E
− |L|, Iθ = |L|. (5.117)
From here we get
E = − mk2
2(Ir + Iθ)2
. (5.118)
The frequencies for each degree of freedom are
ωr = ωθ =
1
k
√
−8E3
m
. (5.119)
Again, we would like to express the energy in terms of a single action variable. In this
case, we consider the canonical transformation (φr, φθ, Ir, Iθ) → (φ1, φ2, I1, I2) generated
by
F2(φr, φθ, I1, I2) = (φθ − φr)I1 + φrI2. (5.120)
Two-dimensional attractive Coulomb problem 67
This implies that
E = −mk
2
2I22
, φ2 = φr, (5.121)
and ω1 = 0, ω2 = ωr. We will omit the subscript and refer to I2 and φ2 simply as I and
φ, respectively. Carrying a similar procedure as that used in the previous section, we
find that the generating function S is
S(ψ, I; x) = Ie sinψ + Iψ − Lθ(ψ, I; x), (5.122)
where e is expressed as
√
1− L2/I2. Since the angle variable φ does not appear explicitly,
we need to calculate the averages in the definition of the metric by making the change
of variable of integration
φ = ψ − e sinψ → dφ = (1− e cosψ)dψ, (5.123)
where e also takes values from 0 to 2π in one period of motion. Furthermore, we need to
check if there is a dependence of the parameters in the expression θ(ψ, I; x). To do this,
we only need to compare (5.115) with (5.116) to conclude that
cos θ =
cosψ − e
1− e cosψ
. (5.124)
From this expression, it is clearly seen that θ does not depend on the parameters x, so
we have Gi = pr∂ir. The position and momentum in terms of (ψ, I; x) are
r(ψ, I; x) = − I2
mk
(1− e cosψ), pr(ψ, I; x) =
mk
I
e sinψ
1− e cosψ
. (5.125)
This allows us to calculate the generators,
G1 = −Ie
m
sinψ = −rpr
m
, (5.126)
G2 = −Ie
k
sinψ = −rpr
k
, (5.127)
and finally, the metric,
gij =
I2 − L2
2
(
1
m2
1
mk
1
mk
1
k2
)
. (5.128)
Notice the resemblance of the metrics for the isotropic harmonic oscillator (5.107) and
the attractive Coulomb problem (5.128). In Section 5.6, we will shed light on this issue
through the Kustaanheimo-Stiefel transformation. We remind that the EBK rule [85]
tells us that for the two-dimensional hydrogen atom we must have exactly I = (n−1/2)~
and L = ℓ~, with n = 1, 2, 3, ..., and ℓ = 0,±1,±2, ...,±(n − 1). For the ground state,
this means that the metric tensor is
gij(I = ~/2, L = 0) =
~
2
8
(
1
m2
1
mk
1
mk
1
k2
)
, (5.129)
which differs from (5.112) by a factor of 4.
68 Examples of the generator approach to the classical metric
5.5 Two-dimensional anisotropic harmonic oscillator
In this section we consider the two-dimensional anisotropic harmonic oscillator which
has two degrees of freedom and three parameters. The Hamiltonian is
Ĥ =
p̂2u + p̂2v
2m
+
ku
2
û2 +
kv
2
v̂2, (5.130)
and we take our parameters to be x = {xi} = (m, ku, kv) with i = 1, 2, 3. The angular
momentum of this system is not conserved, so it cannot be reduced to a single degree of
freedom, unless of course, the two spring constants are made equal.
For the quantum calculation, we employ the ground state only for simplicity. The
normalized ground state wave function is just the product of the wave functions for each
coordinate since the system is separable [86],
ψ(u, v; x) =
(√
mku
π~
)1/4(√
mkv
π~
)1/4
exp
(
−
√
mku
2~
u2
)
exp
(
−
√
mkv
2~
v2
)
. (5.131)
The inner product of two functions is now expressed as a double integral,
〈f |g〉 =
∞
∫
−∞
du
∞
∫
−∞
dv f ∗(u, v; x)g(u, v; x). (5.132)
Of course, once again the connection and curvature are zero. On the other hand, the
quantum metric tensor is
g
(0)
ij =
1
32




2
m2
1
mku
1
mkv
1
mku
1
k2u
0
1
mkv
0 1
k2v




. (5.133)
Now, moving to the classical quantities, we know that the Hamiltonian for this system
can be expressed as
H = Iuωu + Ivωv, (5.134)
where Iu and Iv denote the action variable of the respective coordinate, and where
ωu =
√
ku/m and ωv =
√
kv/m. The coordinates and the momenta are easily written
in terms of action-angle variables as
u(φu, Iu; x) =
√
2Iu√
mku
sinφu, v(φv, Iv; x) =
√
2Iv√
mkv
sinφv (5.135)
and
pu(φu, Iu; x) =
√
2Iu
√
mku cosφu, pv(φv,Iv; x) =
√
2Iv
√
mkv cosφv. (5.136)
Mapping of metrics through the Kustaanheimo-Stiefel transformation 69
The generating function is just the sum of the functions for each degree of freedom [76],
S(φu, φv, Iu, Iv) = Iu(φu + sinφu cosφu) + Iv(φv + sinφv cosφv). (5.137)
We see immediately that it does not depend on the parameters, so the generators will
only be Gi = pu∂iu+ pv∂iv. From here, we get the metric tensor
gij =
1
32




I2u+I
2
v
m2
I2u
mku
I2v
mkv
I2u
mku
I2u
k2u
0
I2v
mkv
0 I2v
k2v




. (5.138)
According to the EBK quantization rule, the actions are Iu = (nu + 1/2)~ and Iv =
(nv+1/2)~, where nu, nv = 0, 1, 2, .... Since we are considering the ground state only, the
actions become Iu = Iv = ~/2. Again, we notice a discrepancy between the quantum and
classical results. On a side note, in the case of an isotropic oscillator, i.e., ku = kv = k,
the quantum metric tensor takes the form (cf. (5.74)),
gij(ku = kv = k) =
1
16
(
1
m2
1
mk
1
mk
1
k2
)
, (5.139)
whereas its classical counterpart turns out to be (cf. (5.107))
gij(ku = kv = k) =
I2u + I2v
32
(
1
m2
1
mk
1
mk
1
k2
)
. (5.140)
5.6 Mapping of metrics through the Kustaanheimo-
Stiefel transformation
The Kustaanheimo-Stiefel (KS) transformation allows one to relate the harmonic oscil-
lator and the attractive Coulomb problem [87, 88]. In particular, we utilize the two-
dimensional mapping (also known as the Levi-Civita transformation) to obtain in a
straightforward manner the metric tensor of the Coulomb problem through the trans-
formation rule of a second rank tensor. In this section, the barred parameters belong to
the Coulomb problem.
Let us start with the attractive Coulomb Hamiltonian
H̄ =
p2x + p2y
2m̄
− k̄
√
x2 + y2
, (5.141)
where k̄ > 0.
The KS transformation is defined by
x = u2 − v2, y = 2uv,
dt
ds
= u2 + v2. (5.142)
70 Examples of the generator approach to the classical metric
Notice that besides the transformation of coordinates, there is a time rescaling too. The
momenta transform according to
px =
upu − vpv
2(u2 + v2)
, py =
vpu + upv
2(u2 + v2)
. (5.143)
The energy of the Coulomb problem (an integral of motion) in terms of variables (u, v, s)
is
Ē =
p2u + p2v
8m̄(u2 + v2)
− k̄
u2 + v2
, (5.144)
or, after rearranging,
k̄ =
p2u + p2v
8m̄
− Ē(u2 + v2). (5.145)
In order to have periodic orbits, we restrict ourselves to the bounded case so we take
Ē < 0. The above expression resembles the energy of the two-dimensional isotropic
harmonic oscillator with mass m = 4m̄ and spring constant k = −2Ē. To verify that
this is indeed the appropriate Hamiltonian, we can apply the KS transformation directly
on the equations of motion for (5.141) which will take the form
u′ =
pu
4m̄
, p′u = 2Ēu, (5.146)
v′ =
pv
4m̄
, p′v = 2Ēv. (5.147)
where the prime denotes differentiation with respect to the time s. Therefore, we see
that the transformation yields the Hamiltonian
H =
p2u + p2v
2m
+
k
2
(u2 + v2) = E. (5.148)
So far, we have made the following identifications
m = 4m̄, k = −2Ē, E = k̄. (5.149)
It can easily be shown [87] that the angular momenta of the two systems are related as
L = 2L̄. (5.150)
Furthermore, the action variable is the same for both systems, since
I =
√
m
k
E
2
=
√
4m̄
−2Ē
k̄
2
=
√
−m̄k̄
2
2Ē
= Ī . (5.151)
Thus, the spring constant k is given by
k =
m̄k̄2
Ī2
. (5.152)
Mapping of metrics through the Kustaanheimo-Stiefel transformation 71
Now, we are ready to obtain the metric tensor of the attractive Coulomb problem
using the tensor transformation rule
ḡij =
∂xk
∂x̄i
∂xl
∂x̄j
gkl, (5.153)
or, in matrix form,
Ḡ = ATGA, (5.154)
where A is the Jacobian matrix
A =
(
∂m
∂m̄
∂m
∂k̄
∂k
∂m̄
∂k
∂k̄
)
=
(
4 0
k̄2
Ī2
2m̄k̄
Ī2
)
. (5.155)
Rewriting the metric tensor (5.107) in terms of the barred parameters and carrying out
the matrix product (5.154) we get
ḡij =
Ī2 − L̄2
2
(
1
m̄2
1
m̄k̄
1
m̄k̄
1
k̄2
)
, (5.156)
which is precisely the metric of the Coulomb problem (5.128).

Chapter 6
Examples of the time-dependent
approach to the classical metric
In this chapter, we employ the path integral approach to the quantum geometric tensor
[58] and its classical time-dependent counterpart [68] to analyze some models. We see
once again that the classical quantities contain all, or almost all, the parameter space
structure modulo some identifications on the action variables. Moreover, some advan-
tages of the time-dependent approach are manifest, such as not requiring explicitly the
generating function of the transformation to action-angle variables. The results of this
chapter have been published in [68].
6.1 Symmetric coupled harmonic oscillators
We begin by analyzing two coupled harmonic oscillators with Hamiltonian
Ĥ =
1
2
(p̂21 + p̂22) +
k
2
(q̂21 + q̂22) +
k′
2
(q̂1 − q̂2)
2. (6.1)
This system has been extensively employed to analyze quantum entanglement [89–91]
and as a model toward understanding circuit complexity [26]. We take as our adiabatic
parameters x = {xi} = (k, k′). The corresponding deformation operators are
Ô1 =
∂Ĥ
∂k
=
1
2
(
q̂21 + q̂22
)
, Ô2 =
∂Ĥ
∂k′
=
1
2
(q̂1 − q̂2) . (6.2)
To compute the expectation values that are required, we need to introduce new coordi-
nates (Q̂1, Q̂2) and momenta (P̂1, P̂2) given by
q̂1 =
1√
2
(Q̂1 + Q̂2), q̂2 =
1√
2
(Q̂1 − Q̂2), (6.3)
p̂1 =
1√
2
(P̂1 + P̂2), p̂2 =
1√
2
(P̂1 − P̂2), (6.4)
73
74 Examples of the time-dependent approach to the classical metric
such that the Hamiltonian acquires the diagonal form
H =
1
2
(P̂ 2
1 + P̂ 2
2 ) +
ω2
1
2
Q̂2
1 +
ω2
2
2
Q̂2
2, (6.5)
where the frequencies are
ω2
1 = k, ω2
2 = k + 2k′. (6.6)
Given that the system is now separable, we can cast the position of the diagonal system
in terms of time-dependent annihilation and creation operators b̂a(t) and b̂†a(t) (a = 1, 2)
as
Q̂a(t) =
√
~
2ωa
(
b̂†a(t) + b̂a(t)
)
. (6.7)
This, in turn, allows us to express the operators Ôi(t) in terms of b̂a(t) and b̂†a(t). Taking
into account the time dependence, the operators read b̂a(t) = b̂ae
−iωat, b̂†a(t) = b̂†ae
iωat,
with which the resulting integrands Λ
(0)
ij ≡ 〈Ôi(t1), Ôj(t2)〉0 − 〈Ôi(t1)〉0〈Ôj(t2)〉0 are
Λ
(0)
11 =
~
2
8ω2
1
e−2iω1(t1−t2) +
~
2
8ω2
2
e−2iω2(t1−t2),
Λ
(0)
12 =
~
2
4ω2
2
e−2iω2(t1−t2),
Λ
(0)
22 =
~
2
2ω2
2
e−2iω2(t1−t2). (6.8)
Plugging this in (2.55) and using the prescription (2.63) for the time integrals, we find
the quantum geometric tensor for the ground state (which in this case is equal to the
quantum metric tensor due to the lack of an imaginary part)
g
(0)
ij =
1
32


1
ω4
1
+ 1
ω4
2
2
ω4
2
2
ω4
2
4
ω4
1

 . (6.9)
The metric has a non-vanishing determinant det g
(0)
ij = 1/(256ω4
1ω
4
2). Of course, due
to the reason stated above, the Berry curvature is zero. For a general state |m,n〉,
(m,n = 0, 1, 2, ...), the quantum metric is
g
(m,n)
ij =
1
32


m2+m+1
ω4
1
+ n2+n+1
ω4
2
2(n2+n+1)
ω4
2
2(n2+n+1)
ω4
2
4(n2+n+1)
ω4
2

 . (6.10)
In the classical case, we use the Hamiltonian
H =
1
2
(p21 + p22) +
k
2
(q21 + q22) +
k′
2
(q1 − q2)
2. (6.11)
Symmetric coupled harmonic oscillators 75
A transformation analogous to (6.3) and (6.4) casts the Hamiltonian in the diagonal form
H = 1
2
(P 2
1 + P 2
2 ) +
ω2
1
2
Q2
1 +
ω2
2
2
Q2
2. The solution as a function of time is easily expressed
in terms of normal coordinates as
Q1(t) = Q10 cosω1t+
P10
ω1
sinω1t
P1(t) = P10 cosω1t− ω1Q10 sinω1t
Q2(t) = Q20 cosω2t+
P20
ω2
sinω2t
P2(t) = P20 cosω2t− ω2Q20 sinω2t, (6.12)
where we have taken the initial conditions Q1(0) = Q10, P1(0) = P10, and similarly for
Q2 and P2.
First, we compute the Hannay curvature. The deformations functions Oi = ∂H/∂xi
are written in terms of the normal coordinates as
O1(t) =
1
2
[
P10
(
P10 sin
2 ω1t+ ω1Q10 sin 2ω1t
)
ω2
1
+
P20
(
P20 sin
2 ω2t+ ω2Q20 sin 2ω2t
)
ω2
2
+Q2
10 cos
2 ω1t+Q2
20 cos
2 ω2t
]
,
O2(t) =
(P20 sinω2t+ ω2Q20 cosω2t)
2
ω2
2
. (6.13)
The Poisson bracket of the Oi evaluated at different times turns out to be
{O1(t1),O2(t2)} = −2 sinω2(t1 − t2)[P20 sinω2t1 + ω2Q20 cosω2t1][P20 sinω2t2 + ω2Q20 cosω2t2]
ω3
2
,
(6.14)
where we have taken advantage of their canonical invariance and calculated them with
respect to the initial conditions (Q0, P0) rather than (q0, p0). Since the system is separable
in normal coordinates, we can easily express the initial conditions in terms of initial
action-angle variables as
Q10(φ, I) =
√
2I1
ω1
sinφ10, P10(φ, I) =
√
2ω1I1 cosφ10, (6.15)
and similarly for Q20 and P20. Substituting in (6.14), we find that
{O1(t1),O2(t2)} = −4I2 sinω2(t1 − t2) sin(ω2t1 + φ20) sin(ω2t2 + φ20)
ω2
2
. (6.16)
The next step is to compute the average. Since we have two degrees of freedom, the
torus average of a function f(φ) takes the form
〈f〉0 ≡
1
(2π)2
2π
∫
0
dφ10
2π
∫
0
dφ20 f(φ), (6.17)
76 Examples of the time-dependent approach to the classical metric
thus,
〈{O1(t1),O2(t2)}〉0 = −I2 sin 2ω2(t1 − t2)
ω2
2
. (6.18)
All that remains to compute the unique element of the curvature is to perform the time
integrals. We see that
F12 = − I2
ω2
2
0
∫
−∞
dt1
∞
∫
0
dt2 sin 2ω2(t1 − t2) = 0, (6.19)
due to our regularization prescription (2.63).
We now compute the classical metric. Making use of (6.12), we find the Oi as
functions of time and initial action-angle variables:
O1(t) =
I1 sin
2(ω1t+ φ10)
ω1
+
I2 sin
2(ω2t+ φ20)
ω2
,
O2(t) =
2I2 sin
2(ω2t+ φ20)
ω2
. (6.20)
We begin by evaluating the average of each Oi:
〈O1(t)〉0 =
2π
∫
0
dφ10
2π
∫
0
dφ20
[
I1 sin
2(ω1t+ φ10)
ω1
+
I2 sin
2(ω2t+ φ20)
ω2
]
=
I1ω2 + I2ω1
2ω1ω2
, (6.21)
〈O2(t)〉0 =
2π
∫
0
dφ10
2π
∫
0
dφ20
[
2I2 sin
2(ω2t+ φ20)
ω2
]
=
I2
ω2
. (6.22)
The averages of the products of Oi evaluated at different times are
〈O1(t1)O1(t2)〉0 =
I21ω
2
2 cos 2ω1(t2 − t1) + I22ω
2
1 cos 2ω2(t2 − t1) + 2(I1ω2 + I2ω1)
2
8ω2
1ω
2
2
,
〈O1(t1)O2(t2)〉0 =
I2 [2I1ω2 + 2I2ω1 + I2ω1 cos 2ω2(t2 − t1)]
4ω1ω2
2
,
〈O2(t1)O2(t2)〉0 =
I22 [cos 2ω2(t2 − t1) + 2]
2ω2
2
. (6.23)
Symmetric coupled harmonic oscillators 77
The component g11 is
g11 = −
0
∫
−∞
dt1
∞
∫
0
dt2
[
1
8
(
I21 cos 2ω1(t2 − t1)
ω2
1
+
I22 cos 2ω2(t2 − t1)
ω2
2
)]
=
1
32
(
I21
ω4
1
+
I22
ω4
2
)
, (6.24)
whereas the component g12 is
g12 = −
0
∫
−∞
dt1
∞
∫
0
dt2
[
I22 cos 2ω2(t2 − t1)
4ω2
2
]
=
I22
16ω4
2
. (6.25)
Finally, we find the component g22:
g22 = −
0
∫
−∞
dt1
∞
∫
0
dt2
[
I22 cos 2ω2(t2 − t1)
2ω2
2
]
=
I22
8ω4
2
. (6.26)
The resulting classical metric has the form
gij =
1
32


I21
ω4
1
+
I22
ω4
2
2I22
ω4
2
2I22
ω4
2
4I22
ω4
2

 , (6.27)
whose determinant is
det gij =
I21I
2
2
256ω4
1ω
4
2
, (6.28)
and it is always different from zero. Comparing the classical and quantum metrics, (6.27)
and (6.10) respectively, we see that they are related as
g
(m,n)
ij =
gij
~2
, (6.29)
as long as we make the identifications I21 = (m2 + m + 1)~2 and I22 = (n2 + n + 1)~2.
This confirms relation (4.74).
78 Examples of the time-dependent approach to the classical metric
6.2 Linearly coupled harmonic oscillators
We now present a different system that is also important in the study of entanglement
[92–94]. The Hamiltonian under consideration is
Ĥ =
1
2
(p̂21 + p̂22) +
A
2
q̂21 +
B
2
q̂22 +
C
2
q̂1q̂2, (6.30)
where the parameters are x = {xi} = (A,B,C) with A 6= B. This Hamiltonian is not a
particular case of (6.1), so it must be treated separately.
We begin with the computation of the quantum metric. The deformation operators
associated with the parameter space are
Ô1(t) =
q̂21
2
, Ô2(t) =
q̂22
2
, Ô3(t) =
q̂1q̂2
2
. (6.31)
The first step, as we have seen, is to carry out the diagonalization of the Hamiltonian.
With that purpose, we introduce the transformation
q̂1 = Q̂1 cosα + Q̂2 sinα, q̂2 = −Q̂1 sinα + Q̂2 cosα, (6.32)
p̂1 = P̂1 cosα + P̂2 sinα, p̂2 = −P̂1 sinα + P̂2 cosα), (6.33)
Here, tanα = ǫ
|ǫ|
√
ǫ2 + 1− ǫ with ǫ = B−A
C
. Since tanα ∈ (−1, 1), the angle is restricted
as −π
4
< α < π
4
. Once the transformation is effected, we end up with the Hamiltonian
H =
1
2
(P̂ 2
1 + P̂ 2
2 ) +
ω2
1
2
Q̂2
1 +
ω2
2
2
Q̂2
2, (6.34)
where the normal frequencies are
ω2
1 = A− C
2
tanα, ω2
2 = B +
C
2
tanα. (6.35)
The transformations (6.32) and (6.33) help us treat the three deformation operators at
once by applying the chain rule:
Ôi =
∂Ĥ
∂xi
= ω1Q̂
2
1∂iω1 + ω2Q̂
2
2∂iω2 − (ω2
1 − ω2
2)Q̂1Q̂2∂iα, (6.36)
and from here, we can write the Q̂a in terms of creation and annihilation operators
as in (6.7), but with the frequencies (6.35). The integrands Λ
(0)
ij = 〈Ôi(t1), Ôj(t2)〉0 −
〈Ôi(t1)〉0〈Ôj(t2)〉0 can then be obtained and have the form
Λ
(0)
ij =~
2
[
∂iω1∂jω1
2
e−2iω1(t1−t2)
+
∂iω2∂jω2
2
e−2iω2(t1−t2) + ∂iα∂jα
(ω2
1 − ω2
2)
2
4ω1ω2
e−i(ω1+ω2)(t1−t2)
]
. (6.37)
Linearly coupled harmonic oscillators 79
Substituting this in (2.55), we get the quantum metric tensor
g
(0)
ij =
∂iω1∂jω1
8ω2
1
+
∂iω2∂jω2
8ω2
2
+ ∂iα∂jα
[
1
4
(
ω1
ω2
+
ω2
ω1
)
− 1
2
]
, (6.38)
which explicitly reads
g
(0)
ij =
1
32ω4
1
Mij +
1
32ω4
2
Nij +
Lij
4 (ω2
2 − ω2
1)
2
[
1
4
(
ω1
ω2
+
ω2
ω1
)
− 1
2
]
, (6.39)
where the matrices Mij, Nij and Lij are
Mij =
1
4




(1 + µ)2 ν2 −(1 + µ)ν
ν2 (1− µ)2 −(1− µ)ν
−(1 + µ)ν −(1− µ)ν ν2




,
Nij =
1
4




(1− µ)2 ν2 (1− µ)ν
ν2 (1 + µ)2 (1 + µ)ν
(1− µ)ν (1 + µ)ν ν2




,
Lij =




ν2 −ν2 µν
−ν2 ν2 −µν
µν −µν µ2




, (6.40)
with µ = cos 2α = ǫ√
ǫ2+1
and ν = sin 2α = 1√
ǫ2+1
. Since the quantum geometric tensor
is real, we have a vanishing Berry curvature
F
(0)
ij = 0. (6.41)
For a general state |m,n〉, (m,n = 0, 1, 2, ...), one gets the quantum metric
g
(m,n)
ij =
m2 +m+ 1
32ω4
1
Mij +
n2 + n+ 1
32ω4
2
Nij +
Lij
4 (ω2
2 − ω2
1)
2
[(
ω1
ω2
+
ω2
ω1
)
×
(
m+
1
2
)(
n+
1
2
)
− 1
2
]
, (6.42)
and the Berry curvature
F
(m,n)
ij = 0. (6.43)
We now compute the classical metric and the Hannay curvature. The Hamiltonian
of the system is, naturally,
H =
1
2
(p21 + p22) +
A
2
q21 +
B
2
q22 +
C
2
q1q2.. (6.44)
80 Examples of the time-dependent approach to the classical metric
As in the quantum case, we can use transformations analogous to (6.32) and (6.33) to
diagonalize the Hamiltonian. The deformation functions will then take the form
Oi = ω1Q
2
1∂iω1 + ω2Q
2
2∂iω2 + (ω2
2 − ω2
1)Q1Q2∂iα. (6.45)
The normal coordinates can be expressed in terms of initial action-angle variables (φ0, I)
and time as
Qa(t) =
√
2Ia
ωa
sin(ωat+ φ0a), a = 1, 2 . (6.46)
This allows us to form the integrands Λij = 〈Oi(t1)Oj(t2)〉0 − 〈Oi(t1)〉0〈Oj(t2)〉0 which
turn out to be
Λij =
I1
2
∂iω1∂jω1 cos[2ω1(t1 − t2)] +
I2
2
∂iω2∂jω2 cos[2ω2(t1 − t2)]
+
I1I2
ω1ω2
(ω2
2 − ω2
1)
2∂iα∂jα cos[ω1(t1 − t2)] cos[ω2(t1 − t2)]. (6.47)
Plugging them in (4.75), we obtain the classical metric
gij =
∂iω1∂jω1
8ω2
1
I21 +
∂iω2∂jω2
8ω2
2
I22 + I1I2
(
ω1
ω2
+
ω2
ω1
)
∂iα∂jα, (6.48)
which explicitly reads
gij =
I21
32ω4
1
Mij +
I22
32ω4
2
Nij +
I1I2
4 (ω2
2 − ω2
1)
2
(
ω1
ω2
+
ω2
ω1
)
Lij, (6.49)
where the matrices Mij, Nij and Lij are the same as those of the quantum case (6.40).
We now want to compare the quantum and classical metrics, (6.42) and (6.49), respec-
tively. By using the Bohr-Sommerfeld quantization rule I1 =
(
m+ 1
2
)
~, I2 =
(
n+ 1
2
)
~,
and the identifications I21 = (m2 +m+ 1)~2, I22 = (n2 + n+ 1)~2, we find the relation
g
(m,n)
ij =
1
~2
gij −
Lij
8(ω2
2 − ω2
1)
2
, (6.50)
instead of our semiclassical relation (4.74). We see that the extra term in (6.50) does not
involve the quantum numbers m,n, it only contains parameters and is of order ~
0. The
reason behind this result is the ambiguity of operator ordering in some of the expectation
values. They are
〈Q̂1P̂2Q̂2P̂1〉m,n =
~
2
4
, 〈Q̂2P̂1Q̂1P̂2〉m,n =
~
2
4
, (6.51)
whereas the corresponding averages in the classical setting are
〈Q1P2Q2P1〉0 = 0, 〈Q2P1Q1P2〉0 = 0. (6.52)
Singular Euclidean oscillator 81
Finally, we compute the Hannay curvature. We need the non-equal-time Poisson
brackets, which we choose to calculate in terms of the initial normal coordinates (Q0, P0):
〈{Oi(t1),Oj(t2)}〉0 =− I1 cos[ω1(t1 − t2)]
(
4∂iω1∂jω1 sin[ω1(t1 − t2)]
+ ∂iα∂jα
(ω2
1 − ω2
2)
2
ω1ω2
sin[ω2(t1 − t2)]
)
− I2 cos[ω2(t1 − t2)]
(
4∂iω2∂jω2 sin[ω2(.t1 − t2)]
+ ∂iα∂jα
(ω2
1 − ω2
2)
2
ω1ω2
sin[ω1(t1 − t2)]
)
. (6.53)
Substituting this in (4.77), we finally get that the Hannay curvature is zero, which
coincides with its quantum counterpart.
6.3 Singular Euclidean oscillator
We now present the singular Euclidean oscillator. This model has been used in the
study of quantum rings, which are semiconductor ring-shaped systems [95, 96]. The
Hamiltonian of this oscillator is
H =
p2
2
+
α2
2r2
+
ω2r2
2
. (6.54)
We first solve for the ground state and then compute the quantum metric tensor. After
that, we set out to solve the corresponding classical system and find the classical metric.
At the end, we compare both results and establish some prescriptions for the action
variables.
To solve the system, we shall restrict ourselves to the two-dimensional case, so we
can introduce a polar coordinate system as
x = r cos θ, y = r sin θ. (6.55)
The Hamiltonian then reads
H =
p2r
2
+
p2θ + α2
2r2
+
ω2r2
2
, (6.56)
where we clearly see that θ is a cyclic coordinate and hence its conjugate momentum
pθ is a constant. The time-independent Schrödinger equation for the singular Euclidean
oscillator is
∂2ψ
∂r2
+
1
r
∂ψ
∂r
+
1
r2
∂2ψ
∂θ2
+
2
~2
[
E −
(
α2
2r2
+
ω2r2
2
)]
ψ = 0. (6.57)
82 Examples of the time-dependent approach to the classical metric
We can easily separate the angular part by substituting ψ(r, θ) = R(r)Θ(θ). This results
in
Θ(θ) =
eiℓθ√
2π
, ℓ = 0,±1,±2, ... , (6.58)
and the radial equation
d2R
dr2
+
1
r
dR
dr
+
2
~2
[
E −
(
~
2ℓ2 + α2
2r2
+
ω2r2
2
)]
R = 0. (6.59)
The normalization condition on the wave function is
〈ψ|ψ〉 =
2π
∫
0
dθ |Θ(θ)|2
∞
∫
0
dr r|R(r)|2 = 1. (6.60)
As is usual, we require both the angular and the radial wave functions to be normalized.
We see from (6.58) that the angular wave function automatically fulfills this condition,
so we are only left effectively with the radial wave function. This means that our inner
products will be of the form
〈f |g〉 =
∞
∫
0
dr rf ∗(r)g(r). (6.61)
Since we are only interested in the ground state, we set ℓ = 0. From (6.59) we expect
that when r → ∞, we can recover the harmonic oscillator wave function. To account for
the behavior for all r, we make the following ansatz
R0(r) = Nrβe−
ω
2~
r2 , (6.62)
where N is a normalization constant, and β is a parameter we must determine. We plug
our ansatz in (6.59), which leads to
− α2 + ~
2β2 + 2r2[E0 − (β + 1)ω~] = 0. (6.63)
Equating each coefficient of a power of r to zero, we find that
β = α/~, E0 = (α + ~)ω. (6.64)
We only need to find the constant N by solving the condition
∞
∫
0
dr rR2
0(r) = N2
∞
∫
0
dr r1+2βe−
ωr2
~ = 1. (6.65)
We obtain
N =
√
2ω
αΓ
(
α
~
)
(ω
~
) α
2~
, (6.66)
Singular Euclidean oscillator 83
where Γ(z) is the gamma function [97]
Γ(z) =
∞
∫
0
dt tz−1e−t, Rez > 0. (6.67)
Hence, the radial ground state wave function reads
R0(r) =
√
2ω
αΓ
(
α
~
)
(ω
~
) α
2~
r
α
~ e−
ω
2~
r2 , (6.68)
which has the energy
E0 = (α + ~)ω. (6.69)
We can readily corroborate that these results reduce to the two-dimensional harmonic
oscillator (see [98]) when we take the limit α → 0:
E0(α = 0) = ~ω, R0(r;α = 0) =
√
2ω
~
e−
ω
2~
r2 . (6.70)
Now that we have the solution, we proceed to compute the quantum metric tensor
using (2.20):
g
(0)
ij (x) = Re (〈∂iψ0|∂jψ0〉 − 〈∂iψ0|ψ0〉〈ψ0|∂jψ0〉) . (6.71)
We take as our parameters x = {xi} = (ω, α), i = 1, 2. The fact that we have a real
wave function implies that the Berry connection (and, of course, the Berry curvature) is
zero [4]; hence, the second term vanishes, and the metric tensor simplifies to
g
(0)
ij (x) = 〈∂iψ0|∂jψ0〉. (6.72)
Plugging the wave function (6.68) in this expression, we find the following components:
g
(0)
11 =
α + ~
4~ω2
,
g
(0)
12 = − 1
4~ω
,
g
(0)
22 =
ψ1(1 + α/~)
4~2
, (6.73)
where ψ1(z) is the trigamma function defined as [97]
ψ1(z) =
d2
dz2
ln Γ(z), (6.74)
which has the series representation
ψ1(z) =
∞
∑
n=0
1
(z + n)2
. (6.75)
84 Examples of the time-dependent approach to the classical metric
The determinant of the quantum metric tensor is
det g
(0)
ij =
(
1 + α
~
)
ψ1
(
1 + α
~
)
− 1
16~2ω2
, (6.76)
and it is different from zero for any α. By taking the limit, we can see that it only
approaches zero when α → ∞.
In the classical setting, we consider the Hamiltonian
H =
p2r
2
+
p2θ + α2
2r2
+
ω2r2
2
, (6.77)
and try to find the classical analog of the quantum metric tensor and the Berry curvature
proposed in [67]. We note that in order to have orbits with a fixed energy E, the condition
E > ω
√
p2θ + α2 needs to be fulfilled. This guarantees that the energy is greater than
the minimum of the effective radial potential. Due to the presence of the parameter α,
the Runge-Lenz vector will not be a constant of motion, and hence, the orbit will precess
[76]. The action variables of the singular Euclidean oscillator are [95]
Iθ =
1
2π
∮
dθ pθ = pθ,
Ir =
1
2π
∮
dr pr =
E
2ω
− p̃θ
2
, (6.78)
where p̃θ ≡
√
p2θ + α2 =
√
I2θ + α2. The condition for the orbits is then E > ωp̃θ. From
these expressions, we see that the energy in terms of action variables can be written as
E = ω
(
2Ir +
√
I2θ + α2
)
. (6.79)
Given that pθ is a constant of the motion, the generating function that solves the time-
independent Hamilton-Jacobi equation is given by [76]
S(r, θ, E, pθ) = pθθ +
∫
dr pr. (6.80)
From here, we get the angle variables [95]
φθ =
(
∂S
∂pθ
)
r,θ,E
+
(
∂S
∂E
)
r,θ,pθ
(
∂E
∂pθ
)
Ir
= θ − pθ
2p̃θ
arcsin
(p̃θ + ωr2)
√
2Er2 − p̃2θ − ω2r4
(E + ωp̃θ)r2
,
φr =
(
∂S
∂E
)
r,θ,pθ
(
∂E
∂Ir
)
pθ
= − arcsin
E − ω2r2
√
E2 − ω2p̃2θ
. (6.81)
We notice that we can solve for r from the second equation:
r =
(
E +
√
E2 − ω2p̃2θ sinφr
)1/2
ω
. (6.82)
Singular Euclidean oscillator 85
From the definition of the angle variable, we know that φr = ωrt − β, where ωr =
∂E/∂Ir = 2ω, and β is a constant related to the initial position. With this result at
hand, we can easily write the radial coordinate r and its conjugate momentum pr = ṙ as
r(t) =
[
E +
√
E2 − ω2p̃2θ sin(2ωt− β)
]1/2
ω
,
pr(t) =
√
E2 − ω2p̃2θ cos(2ωt− β)
[
E +
√
E2 − ω2p̃2θ sin(2ωt− β)
]1/2
. (6.83)
The next step is to use the initial conditions to solve for β. Setting r(0) = r0 and
pr(0) = pr0, we can solve for the sine and cosine:
cos β =
ωr0pr0
√
E2 − ω2p̃2θ
, sin β =
E − ω2r20
√
E2 − ω2p̃2θ
. (6.84)
Now, we find the explicit form of the deformation functions Oi expanding the trigonomet-
ric functions in (6.83) and substituting the above relations. Recall that the parameters
we chose are x = {xi} = (ω, α). Thus,
O1 =
∂H
∂ω
= ωr2 =
E + ωr0pr0 sin 2ωt− (E − ω2r20) cos 2ωt
ω
,
O2 =
∂H
∂α
=
α
r2
=
αω2
E + ωr0pr0 sin 2ωt− (E − ω2r20) cos 2ωt
. (6.85)
We still need to write the initial conditions, r0 and pr0, in terms of initial action-angle
variables (φr0, φθ0, Ir0, Iθ0) as
r0 =
[
E +
√
E2 − ω2p̃2θ sinφr0
]1/2
ω
, pr0 =
√
E2 − ω2p̃2θ cosφr0
[
E +
√
E2 − ω2p̃2θ sinφr0
]1/2
, (6.86)
where E is given by (6.79). Finally, we substitute these expressions in (6.85) to find that
O1(t) =
E +
√
E2 − ω2p̃2θ sin(φr0 + 2ωt)
ω
,
O2(t) =
αω2
E +
√
E2 − ω2p̃2θ sin(φr0 + 2ωt)
. (6.87)
Notice that, despite its appearance, O2(t) is continuous everywhere, since the denomi-
nator is never zero. Introducing the parameter a =
√
1− ω2p̃2θ/E
2, which has the range
0 < a < 1, we can recast (6.87) in the form
O1(t) =
E
ω
[1 + a sin(φr0 + 2ωt)] ,
O2(t) =
αω2
E
1
1 + a sin(φr0 + 2ωt)
. (6.88)
86 Examples of the time-dependent approach to the classical metric
The formula (4.77) for the Hannay curvature requires the evaluation of the Poisson
bracket {O1(t1),O2(t2)} with respect to the initial conditions. We take advantage of the
canonical invariance of the bracket and choose to evaluate it in terms of action-angle
variables, i.e.,
{O1(t1),O2(t2)} =
∂O1(t1)
∂φr0
∂O2(t2)
∂Ir0
− ∂O1(t1)
∂Ir0
∂O2(t2)
∂φr0
. (6.89)
Since the Oi(t) do not depend on φθ0, we omitted the derivatives with respect to it in
this expression. Now, given (6.87) we can compute the Poisson bracket taking care of
including the dependence on Ir of E and a. We obtain
{O1(t1),O2(t2)} =
4αω2 sinω(t1 − t2)
E
{
cosω(t1 − t2) + a sin[ω(t1 + t2) + φr0]
[1 + a sin(φr0 + 2ωt2)]
2
}
.
(6.90)
The torus average of a function f(φ0) for the two degrees of freedom, φr0 and φθ0, is
〈f〉0 =
1
(2π)2
2π
∫
0
dφr0
2π
∫
0
dφθ0 f(φ0). (6.91)
However, the functions Oi(t) do not depend on φθ0, thus, the average is taken only with
respect to φr0. The average of the Poisson bracket, then, is
〈{O1(t1),O2(t2)}〉0 =
2αω2 sin 2ω(t1 − t2)
E
√
1− a2
=
2αω
p̃θ
sin 2ω(t1 − t2). (6.92)
The final step is to integrate this expression with respect to both t1 and t2. The result
is easily found using the regularization prescription (2.63). We have that
F12 =
2αω
p̃θ
0
∫
−∞
dt1
∞
∫
0
dt2 sin 2ω(t1 − t2) = 0, (6.93)
just as in the quantum case.
To compute the quantum metric tensor, let us evaluate the torus averages. We begin
with the average of each Oi:
〈O1(t)〉0 =
1
2π
2π
∫
0
dφr0 O1(t) =
E
2πω
2π
∫
0
dφr0 [1 + a sin(φr0 + 2ωt)] =
E
ω
, (6.94)
〈O2(t)〉0 =
1
2π
2π
∫
0
dφr0 O2(t) =
αω2
2πE
2π
∫
0
dφr0
1 + a sin(φr0 + 2ωt)
=
αω
p̃θ
. (6.95)
Singular Euclidean oscillator 87
Now, we compute the average of the product of Oi evaluated at different times:
〈O1(t1)O1(t2)〉0 =
1
2π
2π
∫
0
dφr0 O1(t1)O1(t2)
=
E2
2πω2
2π
∫
0
dφr0 [1 + a sin(φr0 + 2ωt1)] [1 + a sin(φr0 + 2ωt2)]
=
E2
ω2
[
1 +
a2
2
cos 2ω(t2 − t1)
]
=
E2
ω2
+
E2 − ω2p̃2θ
2ω2
cos 2ω(t2 − t1). (6.96)
〈O1(t1)O2(t2)〉0 =
1
2π
2π
∫
0
dφr0 O1(t1)O2(t2)
=
αω
2π
2π
∫
0
dφr0
1 + a sin(φr0 + 2ωt1)
1 + a sin(φr0 + 2ωt2)
= αω
[
1− 1√
1− a2
cos 2ω(t2 − t1)
]
=
αE
p̃θ
− α
p̃θ
(E − ωp̃θ) cos 2ω(t2 − t1). (6.97)
〈O2(t1)O2(t2)〉0 =
1
2π
2π
∫
0
dφr0 O2(t1)O2(t2)
=
α2ω4
2πE2
2π
∫
0
dφr0
[1 + a sin(φr0 + 2ωt1)] [1 + a sin(φr0 + 2ωt2)]
=
α2ω4
E2
[
1√
1− a2
1
1− a2 cos2 ω(t2 − t1)
]
=
α2ω3
Ep̃θ
1
1− a2 cos2 ω(t2 − t1)
. (6.98)
88 Examples of the time-dependent approach to the classical metric
With these results, we can start calculating the classical metric. The component g11 is
g11 = −
0
∫
−∞
dt1
∞
∫
0
dt2
[
E2
ω2
+
E2 − ω2p̃2θ
2ω2
cos 2ω(t2 − t1)−
E2
ω2
]
= −E
2 − ω2p̃2θ
2ω2
0
∫
−∞
dt1
∞
∫
0
dt2 cos 2ω(t2 − t1)
= −E
2 − ω2p̃2θ
2ω2
(
− 1
4ω2
)
=
E2 − ω2p̃2θ
8ω4
=
I2r + Ir
√
I2θ + α2
2ω2
, (6.99)
whereas the component g12 is
g12 = −
0
∫
−∞
dt1
∞
∫
0
dt2
[
αE
p̃θ
− α
p̃θ
(E − ωp̃θ) cos 2ω(t2 − t1)−
αE
p̃θ
]
=
α
p̃θ
(E − ωp̃θ)
0
∫
−∞
dt1
∞
∫
0
dt2 cos 2ω(t2 − t1)
=
α
p̃θ
(E − ωp̃θ)
(
− 1
4ω2
)
= − α
4ω2p̃θ
(E − ωp̃θ)
=
−αIr
2ω
√
I2θ + α2
. (6.100)
We see that in both calculations the divergent terms cancel, analogous to the different
examples that appear in [58]. Now, the component g22 is
g22 = −
0
∫
−∞
dt1
∞
∫
0
dt2
[
α2ω3
Ep̃θ
1
1− a2 cos2 ω(t2 − t1)
− α2ω2
p̃2θ
]
=
α2ω2
p̃2θ
0
∫
−∞
dt1
∞
∫
0
dt2
[
1−
√
1− a2
1− a2 cos2 ω(t2 − t1)
]
, (6.101)
where we have left the function inside the brackets in terms of a for easier manipulation.
Given the complicated form of the integral and that the divergent terms do not seem to
Singular Euclidean oscillator 89
be canceled, we try an expansion in Fourier series on the second term so that we have
to integrate only trigonometric functions as the prescription (2.63) indicates. Defining
T ≡ t2 − t1 and the function f(T ) as
f(T ) ≡ 1
1− a2 cos2 ωT
, (6.102)
we Fourier expand it
f(T ) =
c0
2
+
∞
∑
n=1
cn cos
(
2nπT
P
)
, (6.103)
where
cn =
2
P
P
∫
0
dT f(T ) cos
(
2nπT
P
)
, n = 0, 1, 2, ... , (6.104)
and P is such that f(T + P ) = f(T ). Notice that the function f(T ) is not singular
because of the condition 0 < a < 1. Since we are dealing with an even function of T ,
there is no need to include the sine terms in the series. The period P is easily seen to
be π/ω, hence, the expression for the nth coefficient of the series is
cn =
2ω
π
π/ω
∫
0
dT
cos 2nωT
1− a2 cos2 ωT
. (6.105)
Here are listed the first three Fourier coefficients:
c0 =
2√
1− a2
,
c1 = − 2√
1− a2
+
4
a2
√
1− a2
− 4
a2
,
c2 = −16
a4
+
2√
1− a2
− 16
a2
√
1− a2
+
8
a2
+
16
a4
√
1− a2
. (6.106)
With the Fourier expansion, the component g22 of the classical metric is
g22 =
α2ω2
p̃2θ
0
∫
−∞
dt1
∞
∫
0
dt2
[
1−
√
1− a2
(
1√
1− a2
+
∞
∑
n=1
cn cos 2nωT
)]
= −α
2ω2
p̃2θ
√
1− a2
∞
∑
n=1
cn
0
∫
−∞
dt1
∞
∫
0
dt2 cos 2nω(t2 − t1)
=
α2
4p̃2θ
√
1− a2
∞
∑
n=1
cn
n2
=
α2ω
4p̃θE
∞
∑
n=1
cn
n2
. (6.107)
90 Examples of the time-dependent approach to the classical metric
Here, it is evident how the divergent terms cancel each other. A closed expression for g22
can be given provided that we use the expression for the nth Fourier coefficient (6.105)
and interchange the sum and integral symbols:
g22 =
α2ω2
2πp̃θE
π/ω
∫
0
dT
( ∞
∑
n=1
cos 2nωT
n2
1
1− a2 cos2 ωT
)
. (6.108)
Noting that the quadratic Bernoulli polynomial, B2(x) = x2 − x + 1
6
, has the series
representation [99]
B2(x) =
1
π2
∞
∑
n=1
cos 2nπx
n2
, 0 ≤ x ≤ 1, (6.109)
we can set x = ωT/π to get
∞
∑
n=1
cos 2nωT
n2
= π
(
π
6
− ωT +
ω2T 2
π
)
, (6.110)
which, upon substitution into (6.108), yields
g22 =
α2ω2
2p̃θE
(
π
6
I1 − ωI2 +
ω2
π
I3
)
. (6.111)
The three integrals that appear are
I1 =
π/ω
∫
0
dT
1
1− a2 cos2 ωT
,
I2 =
π/ω
∫
0
dT
T
1− a2 cos2 ωT
,
I3 =
π/ω
∫
0
dT
T 2
1− a2 cos2 ωT
. (6.112)
The first and the second have the simple results
I1 =
π
ω
√
1− a2
=
πE
ω2p̃θ
, I2 =
π2
2ω2
√
1− a2
=
π2E
2ω3p̃θ
, (6.113)
whereas the third one turns out to be
I3 =
π
ω3
√
1− a2
[
π2
3
+ Li2β
]
, (6.114)
Singular Euclidean oscillator 91
with β ≡ 2−a2−2
√
1−a2
a2
bounded between 0 and 1. The function Li2(z) is the dilogarithm
defined as [99]
Li2(z) =
0
∫
z
dt
ln(1− t)
t
, (6.115)
or, equivalently,
Li2(z) =
∞
∑
n=1
zn
n2
, |z| < 1. (6.116)
Substitution of a gives
I3 =
πE
ω4p̃θ
[
π2
3
+ Li2
(
E − ωp̃θ
E + ωp̃θ
)]
. (6.117)
Thus, the metric component g22 simplifies to
g22 =
α2
2p̃2θ
Li2
(
Ir
Ir + p̃θ
)
. (6.118)
The determinant of the classical metric is
det gij =
α2I2r
4ω2p̃2θ
[(
1 +
p̃θ
Ir
)
Li2
(
Ir
Ir + p̃θ
)
− 1
]
. (6.119)
We now set out to compare the quantum and classical metrics. It is easily verified
that the components g11 and g12 of both metrics follow the rule gij = ~
2g
(0)
ij provided
that we use the quantization conditions
Ir =
~
2
, I2r =
~
2
2
, Iθ = 0. (6.120)
Regarding the component g22 of the classical metric, we expand it in a Taylor series in
Ir and find
g22 =
Ir
2α
− 3I2r
8α2
+
11I3r
36α3
− 25I4r
96α4
+
137I5r
600α5
− 49I6r
240α6
+O
(
I7r
)
. (6.121)
In order to contrast this result with the quantum computation, we try an expansion on
the trigamma function that appears in the element g(0)22 . We first define the variable
z ≡ α/~, and employ the recurrence relation
ψ1(1 + z) = ψ1(z)−
1
z2
. (6.122)
Now, we make use of the asymptotic expansion of the trigamma function [97]:
ψ1(z) ≈
1
z
+
1
2z2
+
1
6z3
− 1
30z5
+
1
42z7
+O
(
1
z9
)
, (6.123)
92 Examples of the time-dependent approach to the classical metric
which is valid when z → ∞, or, equivalently, when ~/α → 0. This yields
~
2g
(0)
22 ≈ ~
4α
− ~
2
8α2
+
~
3
24α3
− ~
5
120α5
+O(~7). (6.124)
If we want to match the classical and quantum results, we should set the following
quantization rules:
Ir/~ = 1/2, I2r /~
2 = 1/3, I3r /~
3 = 3/22, I4r /~
4 = 0, I5r /~
5 = − 5
137
. (6.125)
Notice that, analogous to the quartic anharmonic oscillator that appears in [67], we
have different quantization conditions for every power of the action variable. This only
indicates that in order to get a better concordance, the semiclassical approximation must
get further contributions from superior powers of ~. Another possibility to compare the
quantum and classical results is to take Ir = ~/2, which is suggested by the matching of
the other metric components. Using this condition, we get
g22(Ir = ~/2, Iθ = 0) =
~
4α
− 3~2
32α2
+
11~3
288α3
− 25~4
1536α4
+
137~5
19200α5
− 49~6
15360α6
+O(~7).
(6.126)
We can see that the first term is in perfect agreement with the quantum expression
(6.124). In Table 6.1, subsequent terms corresponding to higher powers of ~ are shown
for both the classical and quantum case.
Power of ~ Classical g22 Quantum g
(0)
22
2 − 3
32
≈ −0.09375 −1
8
= −0.125
3 11
288
≈ 0.03819 1
24
≈ 0.04166
4 − 25
1536
≈ −0.0162 0
Table 6.1: Comparison of the classical and quantum g22 components of the metric. In the classical
case, we have taken Ir = ~/2.
6.4 Spin-half particle in an external magnetic field
We now consider a spin 1/2 particle in a magnetic field which varies adiabatically. This
system was studied by Berry [6], who found the famous result that the geometrical phase
is proportional to the solid angle subtended by the curve that the magnetic field vector
traces in space during its adiabatic evolution. We would like now to address the classical
counterpart of this system. To do so, we follow the analysis of Gozzi, Thacker and
Rohrlich [100, 101], and from there, we compute the classical metric and the Hannay
Spin-half particle in an external magnetic field 93
curvature. We work with two complex Grassmann variables such that ψaψb + ψbψa = 0,
(a, b = 1, 2) to treat the spin from a classical standpoint (for details, see Appendix B).
The Lagrangian of the systems is
L = iψ†ψ̇ − ψ†M(B)ψ, (6.127)
where ψ = (ψ1, ψ2)
T , B = (B1, B2, B3) and
M(B) =
3
∑
i=1
Biσ
i =
(
B3 B1 − iB2
B1 + iB2 −B3
)
, (6.128)
with σi the Pauli matrices. To build the associated Hamiltonian, we must find the
momenta Πa. We define them using the right derivative as
Πa = L
←
∂
∂ψ̇a
= iψ∗a, (6.129)
so that the Hamiltonian is
H = Π1ψ̇1 +Π2ψ̇2 − L = ψ†M(B)ψ. (6.130)
Hence, in expanded form, the Hamiltonian reads
H = B3ψ
∗
1ψ1 + (B1 − iB2)ψ
∗
1ψ2 + (B1 + iB2)ψ
∗
2ψ1 − B3ψ
∗
2ψ2. (6.131)
The Poisson brackets of two functions f(ψ, ψ∗) and g(ψ, ψ∗) are defined as
{f, g} = i
2
∑
a=1

f
←
∂
∂ψ∗a
→
∂
∂ψa
g + f
←
∂
∂ψa
→
∂
∂ψ∗a
g

 , (6.132)
which leads to the following fundamental brackets:
{ψa, ψb} = {ψ∗a, ψ∗b} = 0, {ψa, ψ∗b} = iδab. (6.133)
To compute the classical metric, we take as our adiabatic parameters the three com-
ponents of the magnetic field, i.e., x = {xi} = (B1, B2, B3). First, we need the functions
Oi(t), which are easily found to be
Oi(t) =
(
∂H
∂Bi
)
ψ†,ψ
=
∑
a,b
ψ∗a(t)σ
i
abψb(t) = ψ†(t)σiψ(t). (6.134)
Since the Hamiltonian couples both degrees of freedom, it is necessary to find the normal
modes by diagonalizing M through a unitary matrix U given by
U =


B1−iB2√
2B(B−B3)
B1−iB2√
2B(B+B3)√
B−B3√
2B
−
√
B+B3√
2B

 . (6.135)
94 Examples of the time-dependent approach to the classical metric
This yields
M̃ = U †MU =
(
Ω1 0
0 Ω2
)
=
(
B 0
0 −B
)
, (6.136)
where B =
√
B2
1 +B2
2 +B2
3 . The diagonalization procedure induces the transformation
ψ = Uψ̃, so that the transformed Lagrangian and Hamiltonian now read
L = i(ψ̃∗1
˙̃ψ1 + ψ̃∗2
˙̃ψ2)− Ω1ψ̃
∗
1ψ̃1 − Ω2ψ̃
∗
2ψ̃2, (6.137)
H = Ω1ψ̃
∗
1ψ̃1 + Ω2ψ̃
∗
2ψ̃2. (6.138)
The momenta of the normal modes are
Π̃a = L
←
∂
∂ ˙̃ψa
= iψ̃∗a, (6.139)
and the equations of motion turn out to be
˙̃ψa = {H, ψ̃a} = −iΩaψ̃a,
˙̃ψ∗a = {H, ψ̃∗a} = iΩaψ̃
∗
a, (6.140)
which possess the solutions
ψ̃a(t) = ψ̃a(0)e
−iΩat, ψ̃∗a(t) = ψ̃∗a(0)e
iΩat. (6.141)
Thus, the action variables are
Ia =
1
2π
∮
dtΠ̃a(t)
˙̃ψa(t) = ψ̃∗a(0)ψ̃a(0), (6.142)
and the Hamiltonian takes the simple form
H = Ω1I1 + Ω2I2 = B(I1 − I2). (6.143)
We can find the angle variables as
φ̇a =
∂H
∂Ia
= Ωa, (6.144)
which implies that
φa(t) = Ωat+ φa0, (6.145)
where φa0 ≡ φa(0). Hence, the solutions in terms of initial angle variables and time are
ψ̃a(t) = ψ̃a(0)e
−iφa0e−iΩat, (6.146)
and their corresponding complex conjugate. They allow us to express the Oi(t) in terms
of normal coordinates as
Oi(t) = ψ†(t)σiψ(t) = ψ̃†(t)U †σiUψ̃(t) = ψ̃†(t)σ̃iψ̃(t), (6.147)
Spin-half particle in an external magnetic field 95
where we have defined σ̃i ≡ U †σiU . Substituting (6.146), we find the Oi in terms of
initial conditions as
Oi(t) =
∑
a,b
ψ̃∗a(0)ψ̃b(0)e
i(Ωa−Ωb)tei(φa0−φb0)σ̃iab. (6.148)
The σ̃i are
σ̃1 =


B1
B
−B1B3+iB2B
B
√
B2
1+B
2
2
−B1B3−iB2B
B
√
B2
1+B
2
2
−B1
B

 , σ̃2 =


B2
B
−B2B3−iB1B
B
√
B2
1+B
2
2
−B2B3+iB1B
B
√
B2
1+B
2
2
−B2
B

 ,
σ̃3 =


B3
B
√
B2
1+B
2
2
B√
B2
1+B
2
2
B
−B3
B

 . (6.149)
We now calculate the average of Oi(t):
〈Oi(t)〉0 =
∑
a,b
ψ̃∗a(0)ψ̃b(0)e
i(Ωa−Ωb)tσ̃iab〈ei(φa0−φb0)〉. (6.150)
Since 〈ei(φa0−φb0)〉0 = δab, we have that
〈Oi(t)〉0 =
∑
a
ψ̃∗a(0)ψ̃a(0)σ̃
i
aa =
∑
a
Iaσ̃
i
aa. (6.151)
The remaining average is 〈Oi(t1)Oj(t2)〉0. We find
〈Oi(t1)Oj(t2)〉0 =
∑
a,b,c,d
ψ̃∗a(0)ψ̃b(0)ψ̃
∗
c (0)ψ̃d(0)e
i[(Ωa−Ωb)t1+(Ωc−Ωd)t2]σ̃iabσ̃
j
cd〈ei(φa0−φb0+φc0−φd0)〉.
(6.152)
Taking into account that
〈ei(φa0−φb0+φc0−φd0)〉0 = δabδcd + δa1δb2δc2δd1 + δa2δb1δc1δd2, (6.153)
this simplifies to
〈Oi(t1)Oj(t2)〉0 =
∑
a,c
ψ̃∗a(0)ψ̃a(0)ψ̃
∗
c (0)ψ̃c(0)σ̃
i
aaσ̃
j
cc
+ψ̃∗1(0)ψ̃2(0)ψ̃
∗
2(0)ψ̃1(0)e
i(Ω1−Ω2)(t1−t2)σ̃i12σ̃
j
21
+ψ̃∗2(0)ψ̃1(0)ψ̃
∗
1(0)ψ̃2(0)e
−i(Ω1−Ω2)(t1−t2)σ̃i21σ̃
j
12. (6.154)
Further simplification can be made by realizing that in the second line we can move
ψ̃∗2(0) to the left of ψ̃2(0) at the cost of a minus sign, and similarly with ψ̃∗1(0) in the
third line. This manipulation results in
〈Oi(t1)Oj(t2)〉0 =
∑
a,c
IaIcσ̃
i
aaσ̃
j
cc − I1I2
[
ei(Ω1−Ω2)(t1−t2)σ̃i12σ̃
j
21 + e−i(Ω1−Ω2)(t1−t2)σ̃i21σ̃
j
12
]
.
(6.155)
96 Examples of the time-dependent approach to the classical metric
Hence, the integrands Λij(t1, t2) ≡ 〈Oi(t1)Oj(t2)〉0 − 〈Oi(t1)〉0〈Oj(t2)〉0 reduce to
Λij(t1, t2) = −I1I2
[
ei(Ω1−Ω2)(t1−t2)σ̃i12σ̃
j
21 + e−i(Ω1−Ω2)(t1−t2)σ̃i21σ̃
j
12
]
, (6.156)
which, integrated in time gives
gij = −I1I2
4B2
(
σ̃i12σ̃
j
21 + σ̃i21σ̃
j
12
)
. (6.157)
Expressed in matrix form, the classical metric tensor is
gij = −I1I2
2B4




B2
2 +B2
3 −B1B2 −B1B3
−B1B2 B2
1 +B2
3 −B2B3
−B1B3 −B2B3 B2
1 +B2
2




. (6.158)
In order to compare this result with the quantum metric tensor, we need to remem-
ber that the quantum system is described by a two-dimensional Hilbert space which is
spanned by the eigenstates with spin projections +1/2 and −1/2. The calculations will
not be shown here, but they can be consulted in [4]. The result, in spherical coordinates,
is
g
(+)
ij = g
(−)
ij =
1
4




0 0 0
0 1 0
0 0 sin2 θ




, (6.159)
where θ is the polar angle between the magnetic field B and the z axis. Using the tensor
transformation rule, we find that in Cartesian coordinates, the quantum metric tensor
has the form
g
(+)
ij = g
(−)
ij =
1
4B4




B2
2 +B2
3 −B1B2 −B1B3
−B1B2 B2
1 +B2
3 −B2B3
−B1B3 −B2B3 B2
1 +B2
2




. (6.160)
Notice that the quantum and classical metric tensors are related by
gij = −2I1I2g
(±)
ij . (6.161)
This shows that the classical metric reproduces the parameter structure of its quantum
counterpart. Finally, it is not difficult to verify that the determinant of (6.158) is zero.
We can obtain a non-vanishing determinant fixing one parameter, for instance, B1 =
B10 = const., which has the effect of eliminating the first row and column of (6.158).
Hence, we are left with a reduced metric
gredij (B1 = B10) = −I1I2
2B4
(
B2
10 +B2
3 −B2B3
−B2B3 B2
10 +B2
2
)
, (6.162)
Spin-half particle in an external magnetic field 97
with determinant
det gredij (B1 = B10) =
I21I
2
2B
2
10
4B6
, (6.163)
which is different from zero as long as B10 6= 0.
To compute the Hannay curvature, we recall that the Poisson brackets are invariant
under unitary transformations [101], which allows us to take the bracket with respect to
the normal coordinates as
{Oi(t1),Oj(t2)} = i
∑
a

Oi(t1)
←
∂
∂ψ̃∗a(0)
→
∂
∂ψ̃a(0)
Oj(t2) +Oi(t1)
←
∂
∂ψ̃a(0)
→
∂
∂ψ̃∗a(0)
Oj(t2)

 .
(6.164)
This yields
{Oi(t1),Oj(t2)} = i
∑
a,b,c
ψ̃∗b (0)ψ̃c(0)N
ij
abc(t1, t2)e
i(φb0−φc0), (6.165)
where
N ij
abc(t1, t2) ≡ −ei[(Ωa−Ωc)t1+(Ωb−Ωa)t2]σ̃iacσ̃
j
ba + ei[(Ωb−Ωa)t1+(Ωa−Ωc)t2]σ̃ibaσ̃
j
ac. (6.166)
The average results in
〈{Oi(t1),Oj(t2)}〉0 = i
∑
a,b
Ib
[
−ei(Ωa−Ωb)(t1−t2)σ̃iabσ̃
j
ba + e−i(Ωa−Ωb)(t1−t2)σ̃ibaσ̃
j
ab
]
= i(I1 − I2)
[
ei(Ω1−Ω2)(t1−t2)σ̃i12σ̃
j
21 − e−i(Ω1−Ω2)(t1−t2)σ̃i21σ̃
j
12
]
(6.167)
and, upon integration with respect to t1 and t2, it is found that
Fij =
i
4B2
(I1 − I2)
(
σ̃i21σ̃
j
12 − σ̃i12σ̃
j
21
)
(6.168)
In terms of components, we have
F23 =
(I1 − I2)
2B3
B1, F31 =
(I1 − I2)
2B3
B2, F12 =
(I1 − I2)
2B3
B3, (6.169)
which matches the result that Gozzi and Thacker [100] found using the standard defini-
tion of the Hannay curvature. It is easily seen that the relationship between the classical
and quantum curvatures is
Fij = −(I1 − I2)F
(+)
ij = (I1 − I2)F
(−)
ij . (6.170)

Chapter 7
Application to the Dicke model
In this chapter, we analyze the parameter space geometry of the Dicke model with the
aid of the classical and quantum metrics, as well as their corresponding scalar curva-
tures. This is the first place where the developed geometric tools are applied to the study
of quantum phase transitions. In particular, through the application of the truncated
Holstein-Primakoff transformation, we find that in the thermodynamic limit, the clas-
sical and quantum metrics have the same divergent behavior near the quantum phase
transition, as opposed to their corresponding scalar curvatures which approach the same
value there. We also see that under resonance conditions, both scalar curvatures show a
divergence at the critical point.
The Dicke model [102] is a well-known model in quantum optics. It describes a
collection of N two-level atoms interacting with one mode of a bosonic field inside a
cavity. Its quantum and classical dynamics have been explored [103–105], and it has been
widely studied in the context of quantum and classical chaos [106–111], entanglement
and fidelity-related measures [112–117].
The Hamiltonian of the Dicke model is
Ĥ = ω0Ĵz + ωâ†â+
λ√
N
(↠+ â)(Ĵ+ + Ĵ−), (7.1)
where ω0 is the splitting of the two levels, ω is the frequency of the bosonic mode, λ is
the coupling of the dipole interaction between the field and the atoms, â and ↠are the
creation and annihilation operators of the field, and Ĵz, Ĵ± = Ĵx ± iĴy are the collective
spin operators. Also, we have chosen ~ = 1. We see that the operator Ĵ2 = Ĵ2
x + Ĵ2
y + Ĵ2
z
commutes with the Hamiltonian, which implies that the total pseudospin is conserved
and we can restrict ourselves, as usual, to the consideration of the maximum pseudospin
j = N /2. This has the effect of treating the collection of N two-level atoms as a single
(N + 1)-level system with pseudospin j = N /2 [106, 107]. We are interested in the
thermodynamic limit j → ∞, where the system undergoes a quantum phase transition
at the critical coupling λ = λc ≡
√
ωω0
2
that separates the normal phase, λ < λc, and the
superradiant phase, λ > λc.
99
100 Application to the Dicke model
7.1 Analysis in the thermodynamic limit
7.1.1 Normal phase
To describe the system in the thermodynamic limit, we follow the work of Emary and
Brandes [106, 107]. We first use the Holstein-Primakoff transformation [118]
Ĵ+ = b̂†
√
2j − b̂†b̂, Ĵ− =
√
2j − b̂†b̂ b̂, Ĵz = b̂†b̂− j, (7.2)
which is a way to associate the bosonic operators b̂ and b̂† to the angular momentum
operators Ĵz, Ĵ±. After performing this transformation, the Dicke Hamiltonian takes the
form
Ĥ = −jω0 + ω0b̂
†b̂+ ωâ†â+ λ(↠+ â)

b̂†
√
1− b̂†b̂
2j
+
√
1− b̂†b̂
2j
b̂

 . (7.3)
Next, we expand the square roots and take the limit j → ∞, keeping only the zeroth
order term in 1/j. This leads to the effective Hamiltonian
Ĥn = −jω0 + ω0b̂
†b̂+ ωâ†â+ λ(↠+ â)(b̂† + b̂), (7.4)
which is valid for λ < λc, i.e., the normal phase. It is important to mention that
this approach is valid only in the low energy region where the classical Hamiltonian
corresponding to (7.1) has regular dynamics [110]. The term proportional to j, which
is dominant as j increases, is identified as the ground state energy of the model in the
normal phase. From (7.4), we readily recognize that this Hamiltonian corresponds to
two coupled harmonic oscillators, as can be explicitly seen by applying the operator
transformation
q̂1 =
1√
2ω
(↠+ â), p̂1 = i
√
ω
2
(↠− â),
q̂2 =
1√
2ω0
(b̂† + b̂), p̂2 = i
√
ω0
2
(b̂† − b̂), (7.5)
which casts it in the position-momentum representation as
Ĥn = −jω0 −
(ω + ω0)
2
+
1
2
(
p̂21 + p̂22 + ω2q̂21 + ω2
0 q̂
2
2 + 4λ
√
ωω0 q̂1q̂2
)
. (7.6)
We can uncouple the two oscillators by going to the normal coordinates (Q̂1, Q̂2)
through the transformation
(
q̂1
q̂2
)
=
(
cosαn sinαn
− sinαn cosαn
)(
Q̂1
Q̂2
)
, (7.7)
Analysis in the thermodynamic limit 101
and similarly for the corresponding conjugate normal momenta (P̂1, P̂2). The angle αn
is such that
tan 2αn =
4λ
√
ωω0
ω2
0 − ω2
, (7.8)
with tanαn ∈
(
−π
4
, π
4
)
, and we assume that ω0 6= ω. After performing this transforma-
tion, the Hamiltonian acquires the form
Ĥn = −jω0 −
(ω + ω0)
2
+
1
2
(
P̂ 2
1 + P̂ 2
2 + ε21nQ̂
2
1 + ε22nQ̂
2
2
)
, (7.9)
where the two (squared) normal frequencies are
ε21n =
1
2
[
ω2 + ω2
0 −
√
(ω2 − ω2
0)
2 + 16λ2ωω0
]
,
ε22n =
1
2
[
ω2 + ω2
0 +
√
(ω2 − ω2
0)
2 + 16λ2ωω0
]
. (7.10)
We clearly see that at the critical coupling λc =
√
ωω0
2
, the normal frequency ε1n vanishes
and the system reduces effectively to only one normal mode. This feature signals the
quantum phase transition of the model [106, 107].
7.1.2 Superradiant phase
In the case of the superradiant phase (λ > λc), one can derive an effective Hamiltonian
Ĥs by letting the field and the set of atoms acquire macroscopic occupation numbers;
one way to achieve this is by displacing the bosonic operators that appear in (7.3) and
demanding that the linear terms in â and ↠vanish. After expanding the square roots
and changing to the position-momentum representation, we arrive to the Hamiltonian
for the superradiant phase which reads [106, 107]
Ĥs = −j
(
2λ2
ω
+
ω2
0ω
8λ2
)
−4λ2 + ω2
2ω
+
1
2
(
p̂21 + p̂22 + ω2q̂21 +
16λ4
ω2
q̂22 + 2ωω0 q̂1q̂2
)
. (7.11)
As in the normal phase, we use the transformation (7.7), which casts the Hamiltonian
in the form
Ĥs = −j
(
2λ2
ω
+
ω2
0ω
8λ2
)
− 4λ2 + ω2
2ω
+
1
2
(
P̂ 2
1 + P̂ 2
2 + ε21sQ̂
2
1 + ε22sQ̂
2
2
)
, (7.12)
where now the rotation angle αs is such that
tan 2αs =
2ω3ω0
16λ4 − ω4
, (7.13)
102 Application to the Dicke model
and we have assumed that λ 6= ±ω/2. The two resulting (squared) normal frequencies
are
ε21s =
1
2


16λ4 + ω4
ω2
−
√
(
16λ4 − ω4
ω2
)2
+ 4ω2ω2
0

 ,
ε22s =
1
2


16λ4 + ω4
ω2
+
√
(
16λ4 − ω4
ω2
)2
+ 4ω2ω2
0

 . (7.14)
Notice, once more, that at the critical coupling λ = λc, the frequency ε1s vanishes.
Indeed, it can be easily verified that Ĥn(λc) = Ĥs(λc). Furthermore, looking at the
dominant term of order j in the Hamiltonians (7.6) and (7.11), we can read off the
ground state energy for both phases:
Eg
j
=
{
−ω0, λ < λc
−
(
2λ2
ω
+
ω2
0ω
8λ2
)
, λ > λc
. (7.15)
This normalized ground state energy exhibits a discontinuity in its second derivative
at λ = λc, which is precisely the hallmark of the quantum phase transition in this
model. Interestingly, the main features of the transition were reproduced using only the
quadratic approximation coming from the truncated Holstein-Primakoff transformation.
This is one of the main virtues of this approach, which was precisely exploited by Emary
and Brandes in their remarkable papers [106, 107]. We are also taking advantage of this
simple method as a first step toward the understanding of the underlying geometry of
the parameter space. In reference [119], some shortcomings of the truncated Holstein-
Primakoff transformation are addressed.
7.2 Classical and quantum metric tensors
Our aim is now to compute the classical and quantum metrics for the normal and su-
perradiant phases, and compare them to see how well the classical metric captures the
essential information of the quantum system. After that, we analyze the scalar cur-
vatures of both metrics. We fix ω0 = const. and take as our adiabatic parameters the
frequency ω and the strength of the dipole coupling λ, which results in a two dimensional
parameter manifold with coordinates x = {xi} = (ω, λ), i = 1, 2.
7.2.1 Normal phase
We begin our computation of the classical metric tensor for the normal phase using the
time-dependent approach (4.75). The Hamiltonian of the normal phase is the classical
counterpart of (7.6),
Hn,cl = −jω0 −
(ω + ω0)
2
+
1
2
(
p21 + p22 + ω2q21 + ω2
0q
2
2 +4λ
√
ωω0 q1q2
)
. (7.16)
Classical and quantum metric tensors 103
The deformation functions associated with the parameters are
O1n =
∂Hn,cl
∂ω
= ωq21 + λ
√
ω0
ω
q1q2,
O2n =
∂Hn,cl
∂λ
= 2
√
ωω0 q1q2, (7.17)
where we have ignored the terms that do not depend on (qa, pa) since they would not
contribute to the metric integrands Λij(t1, t2) := 〈Oin(t1)Ojn(t2)〉0−〈Oin(t1)〉0〈Ojn(t2)〉0.
Actually, we can deal with both deformation functions simultaneously as in Section 6.2,
and write them as
Oin(t) = ε1nQ
2
1(t)∂iε1n + ε2nQ
2
2(t)∂iε2n + (ε22n − ε21n)Q1(t)Q2(t)∂iαn, (7.18)
where the Qa(t), (a = 1, 2) are the normal coordinates that uncouple the two harmonic
oscillators through the transformation (7.7), and εan are the normal frequencies (7.10).
The next step is to write the Qa(t) as functions of the initial conditions (Qa0, Pa0) and
time as
Qa(t) = Qa0 cos εant+
Pa0
εan
sin εant, (7.19)
and then, the initial conditions in terms of initial action-angle variables (φa0, Ia) as
Qa0 =
√
2Ia
εan
sinφa0, Pa0 =
√
2Iaεan cosφa0. (7.20)
Now, we use the classical torus average to form the integrands Λij(t1, t2) which turn
out to be
Λij(t1, t2) =
1
2
∂iε1n∂jε1nI1 cos [2ε1n(t1 − t2)] +
1
2
∂iε2n∂jε2nI2 cos [2ε2n(t1 − t2)]
+
∂iαn∂jαn
ε1nε2n
(
ε21n − ε22n
)2
cos [2ε1n(t1 − t2)] cos [2ε2n(t1 − t2)] . (7.21)
With the regularization prescription (2.63), we carry out the time integrals and finally
obtain the classical metric for the normal phase, whose components are
gij =
∂iε1n∂jε1n
8ε21n
I21 +
∂iε2n∂jε2n
8ε22n
I22 + ∂iαn∂jαn
(
ε1n
ε2n
+
ε2n
ε1n
)
I1I2. (7.22)
It is clear from this expression that the appearance of ε1n in the denominator causes
a divergence in the metric components at λc =
√
ωω0
2
, since ε1n vanishes at the critical
coupling (see equation (7.10)); this property of the classical metric signals the quantum
phase transition in the Dicke model.
We now compute the quantum metric tensor (2.56) for the normal phase. From the
Hamiltonian (7.6), we obtain the corresponding deformation operators, which can be
written in compact form as
Ôin(t) = ε1nQ̂
2
1(t)∂iε1n + ε2nQ̂
2
2(t)∂iε2n + (ε22n − ε21n)Q̂1(t)Q̂2(t)∂iαn, (7.23)
104 Application to the Dicke model
where Q̂a(t), (a = 1, 2) are the operators corresponding to the normal modes of the diag-
onal Hamiltonian, which can be written in terms of annihilation and creation operators
as
Q̂a(t) =
1√
2εan
(
b̂†a0e
iεant + b̂a0e
−iεant
)
. (7.24)
With these operators at hand, we can compute the combination 1
2
〈{Ôin(t1), Ôjn(t2)}〉0−
〈Ôin(t1)〉0〈Ôjn(t2)〉0 and then use the integral regularization (2.63) to arrive at the com-
ponents of the quantum metric tensor, which turn out to be
g
(0)
ij =
∂iε1n∂jε1n
8ε21n
+
∂iε2n∂jε2n
8ε22n
+ ∂iαn∂jαn
[
1
4
(
ε1n
ε2n
+
ε2n
ε1n
)
− 1
2
]
. (7.25)
We notice, as with the classical metric, that the frequency ε1n appears in the denominator
in equation (7.25), causing a divergence when λ = λc and signaling the quantum phase
transition in the Dicke model. Moreover, with both metrics at our disposal (equations
(7.22) and (7.25)), we find the relation
g
(0)
ij = gij −
1
2
∂iαn∂jαn, (7.26)
where the identifications I1 = I2 = 1/2 and I21 = I22 = 1 were made. We observe that the
quantum metric (7.25) has an extra parameter-dependent term that does not appear in
its classical analog; this term has been related to an anomaly arising from the ordering
of the operators in the quantum case [68].
7.2.2 Superradiant phase
The treatment of the superradiant phase is analogous to that of the normal phase in
both the classical and quantum settings; the difference lies in the explicit expressions of
the rotation angle (7.13) and the normal frequencies (7.14) in terms of the parameters.
The classical counterpart of the Hamiltonian (7.11) is
Hs,cl = −j
(
2λ2
ω
+
ω2
0ω
8λ2
)
− 4λ2 + ω2
2ω
+
1
2
(
p21 + p22 + ω2q21 +
16λ4
ω2
q22 + 2ωω0 q1q2
)
,
(7.27)
and its deformation functions are
O1s =
∂Hs,cl
∂ω
= ωq21 −
16λ4
ω3
q22 + ω0 q1q2,
O2s =
∂Hs,cl
∂λ
=
32λ3
ω2
q22. (7.28)
By following the same steps as in the normal phase, we arrive at the classical and
quantum metrics, which turn out to be
gij =
∂iε1s∂jε1s
8ε21s
I21 +
∂iε2s∂jε2s
8ε22s
I22 + ∂iαs∂jαs
(
ε1s
ε2s
+
ε2s
ε1s
)
I1I2, (7.29)
Classical and quantum metric tensors 105
and
g
(0)
ij =
∂iε1s∂jε1s
8ε21s
+
∂iε2s∂jε2s
8ε22s
+ ∂iαs∂jαs
[
1
4
(
ε1s
ε2s
+
ε2s
ε1s
)
− 1
2
]
, (7.30)
respectively. We see that these metrics have the same form as those of equations (7.22)
and (7.25), and hence satisfy the relation (7.26), but with the normal frequencies ε1s and
ε2s and the rotation angle αs. Remarkably, due to the presence of ε1s in the denominators
of (7.29) and (7.30), both metrics exhibit a divergence at λ = λc, which reveals the
existence of the quantum phase transition.
To gain more insight into this, in Figure 7.1 (a), (b), and (c) we show the components
of the classical and quantum metrics for both phases and fixed values of ω and ω0. Clearly,
we see that both metrics diverge at λc, which signals the quantum phase transition,
however, the metric component g22 of the classical metric at λ = 0 shows a different
behavior than its quantum counterpart; this can be attributed to the anomalous extra
term that appears in (7.25), making the classical metric more sensitive to the vanishing
of the coupling term in the Hamiltonian (7.6).
The scalar curvatures, computed with the aid of equation (1.20), for both the classical
and quantum metrics are shown in Figure 7.1 (d). We observe some features from the
plots. First, the agreement between them is good in the superradiant phase (λ > λc).
Second, an important difference between them appears in the normal phase (λ < λc):
while in the quantum case the scalar curvature takes a constant value very close to −4,
in the classical case the scalar curvature possesses a minimum around λ = 0.16. Such a
difference can be related to the behavior of the component g22 at that phase. It would
be interesting if an analysis of the exact model, for finite j, presents this behavior, as
will be the case with the anisotropic Lipkin-Meshok-Glick in the next chapter. And
third, in the limit λ → λc both scalar curvatures approach each other and tend to −4.
Notice, however, that for λ = λc they are not defined since the classical and quantum
Hamiltonians used to compute the metrics are not valid at that point. This behavior
of the scalar curvatures when λ → λc suggests that the singularity of the classical and
quantum metrics at the phase transition could be created by the coordinates used. It
is worth mentioning that the scalar curvature of the quantum metric resembles the one
found in reference [17], the only difference being the method used to compute it and an
overall sign due to a convention in formula (1.20).
7.2.3 Metrics under resonance
A special important case of the Dicke model is that of resonance, i.e., when ω = ω0. We
are unaware of previous geometric analyses in this case. In order to treat it, we take
the limits αn → π/4 and ω0 → ω in the classical and quantum metrics corresponding to
the normal phase (equations (7.22) and (7.25)), whereas in the superradiant phase we
only set ω0 → ω in the associated metrics (equations (7.29) and (7.30)). The resulting
metric components in terms of the parameters are greatly simplified, and we find that
in the normal phase, the components g12 and g22 of the classical and quantum metrics
106 Application to the Dicke model
0.2 0.4 0.6 0.8
λ
2
4
6
8
g11
λc
(a)
0.2 0.4 0.6 0.8
λ
-15
-10
-5
g12 λc
(b)
0.2 0.4 0.6 0.8
λ5
10
15
20
25
g22
λc
(c)
0.2 0.4 0.6 0.8
λ
-10
-8
-6
-4
-2
R λc
(d)
Figure 7.1: Metric components and scalar curvature of the classical metric (solid blue) and the
quantum metric (dashed red) as a function of λ when ω0 = 1 and ω = 0.8. All the components show a
divergence at the phase transition (dotted orange) with critical coupling λc = 0.447, whereas the scalar
curvature does not.
perfectly match when the identifications I21 = I22 = 1 and I1 = I2 = 1/2 are used. These
components are
g12 = g
(0)
12 =
λ(4λ2 − 3ω2)
8ω(ω2 − 4λ2)2
, (7.31)
g22 = g
(0)
22 =
4λ2 + ω2
4(ω2 − 4λ2)2
. (7.32)
On the other hand, the g11 components do not match, as we can see below
g11 =
16λ4ω3 − 8λ2ω5 + ω7 + λ2
√
ω2 − 4λ2(8λ4 − 6λ2ω2 + 2ω4)
32λ2ω2 (ω2 − 4λ2)5/2
, (7.33)
g
(0)
11 =
−16λ6 + 48λ4ω2 − 23λ2ω4 + 3ω6 − ω
√
ω2 − 4λ2(4λ4 − 3λ2ω2 + ω4)
16ω2(ω2 − 4λ2)5/2
(
ω +
√
ω2 − 4λ2
) . (7.34)
In the superradiant phase, the classical and quantum metric components are more
complicated and do not match, however, it can be seen that all of them diverge at the
Classical and quantum metric tensors 107
critical coupling λc = ω/2. We show in Figure 7.2 (a), (b), and (c) the components of the
metrics under the resonance condition ω = ω0. We can see that the component g11 has
a divergence at λ = 0. Moreover, we observe that in the normal phase, the components
g12 and g22 of the classical metric are exactly the same as those of the quantum metric,
just as we mentioned earlier. It is also worth noting that both metrics show the same
behavior in the limiting cases λ→ λc and λ→ ∞.
In Figure 7.2 (d), we show the scalar curvatures associated with the classical and
quantum metrics. We notice that the scalar curvature in the classical case presents a
divergence at λ = 0, which is inherited from the g11 metric component. Once again,
we see that the effect of the anomaly is to get rid of that behavior in the quantum
result. Additionally, in contrast to the non resonant analysis, both scalar curvatures
diverge at the quantum phase transition in exactly the same way. There is, however, an
alternative approach to the metrics under the resonance condition. One could set ω0 = ω
in the Hamiltonian from the very beginning, and from there, derive the corresponding
Oi. The resulting deformation operators are different from those we have used and lead
to different expressions for both the classical and quantum metrics as functions of ω and
λ. As a matter of fact, it turns out that both metrics have zero scalar curvature in the
whole range of ω and λ, which is not particularly illuminating. We are unaware of the
physical reason behind this result and consider that it deserves further analysis.
To conclude, we would like to stress the fact that both the classical and quantum
metrics exhibit a divergent behavior at the phase transition for the resonant and non-
resonant cases. This is a remarkable result since it shows that the classical metric can be
used to get a first insight into the information contained in the quantum metric tensor.
On the other hand, according to the (classical and quantum) scalar curvatures, the
metric for the resonant case involve a genuine singularity, whereas in the non-resonant
case there seems to be a spurious singularity. This effect could be a consequence of the
fact that the Holstein-Primakoff approximation fails at the quantum phase transition
[119]. Then, in order to clarify this point it would be valuable to carry out a study of the
parameter space of the Dicke model for finite j using the original Hamiltonian (7.1). This
model has features that make the study for finite j subtle, although some authors have
successfully employed techniques that can be implemented numerically and allow the
exploration of the system in various regimes (including chaotic regions) [105, 108, 109].
In the classical setting, we could explore the mean-field Dicke Hamiltonian constructed
with coherent states using as a first approach the perturbative analysis developed in [67]
to find corrections to the quadratic approximation in the thermodynamic limit. The
results should then be compared with the quantum counterpart using a perturbative
approach to the quantum metric tensor [120]. Besides, the chaotic region could be
approached through the study of the adiabatic gauge potential (which is deeply related
to the quantum metric tensor), since it has been found recently that it serves as a
sensitive measure of quantum chaos [121].
108 Application to the Dicke model
0.2 0.4 0.6 0.8
λ
2
4
6
8
g11
λc
(a)
0.2 0.4 0.6 0.8
λ
-15
-10
-5
g12 λc
(b)
0.2 0.4 0.6 0.8
λ5
10
15
20
25
g22
λc
(c)
0.1 0.2 0.3 0.4 0.5 0.6 0.7
λ
-800
-600
-400
-200
R λc
(d)
Figure 7.2: Metric components and scalar curvature of the classical metric (solid blue) and the
quantum metric (dashed red) for the resonant case as a function of λ when ω = 0.8. All of them show
a divergence at the phase transition (dotted orange) with critical coupling λc = 0.4.
Chapter 8
Application to the anisotropic
Lipkin-Meshkov-Glick model
In this chapter, we study the parameter space geometry of the anisotropic Lipkin-
Meshkov-Glick (LMG) model. Some quantum information concepts like fidelity and
fidelity susceptibility have already been applied to the study of classical and quantum
phase transitions in this model [114, 122–128], but a complete geometrical characteriza-
tion of the parameter space geometry is lacking. To do so, we analyze both the classical
and the quantum metrics as well as their scalar curvatures. First, we consider the ther-
modynamic limit using Bloch coherent states in the classical case and the truncated
Holstein-Primakoff transformation on the quantum side, and see that the classical met-
ric is exactly the same as its quantum counterpart after making an identification of the
action variable. Two of the three metric components show a divergence at the quantum
phase transition and the scalar curvature is also singular there. Nevertheless, we find
that the metric is not invertible in one phase of the system, and as a consequence, the
scalar curvature is not defined in that phase. We finally perform a numerical analysis
for finite sizes of the system that shows the precursors of the quantum phase transition
and allows their characterization as functions of the parameters and of the system’s size.
The LMG model consists of N mutually interacting spin-half particles that are also
affected by a transverse magnetic field. It was first introduced in the context of nu-
clear physics [129–131], and it has been deeply studied through various analytic and
numerical techniques [128, 132–136]. It has also been used as a model for Floquet time
crystals [137], in the study of out-of-time order correlators [117], and to illustrate the
orthogonality catastrophe and its relation to quantum speed limit [138].
The Hamiltonian that we shall consider is [123]
Ĥ = −h
∑
i
σiz −
1
N
∑
i<j
(σixσ
j
x + γσiyσ
j
y), (8.1)
where {σiα}, α = x, y, z are the Pauli spin matrices for the ith spin, h, γ are real pa-
rameters, and we have set ~ = 1. From here, it is customary to define the pseudospin
109
110 Application to the anisotropic Lipkin-Meshkov-Glick model
(collective spin) operators as Ĵα =
∑
i σ
i
α/2 and cast the Hamiltonian into the form
Ĥ = −2hĴz −
1
j
(
Ĵ2
x + γĴ2
y
)
, (8.2)
where, as usual, we restrict ourselves to the maximum pseudospin representation with
j = N /2. We also consider h ≥ 0 and −1 < γ < 1, and analyze the system in the
thermodynamic limit j → ∞ where a quantum phase transition occurs.
8.1 Analysis in the thermodynamic limit
The description of the LMG model when j → ∞ begins by taking the expectation value
of the Hamiltonian (8.2) in spin coherent states |z〉 given by [117, 124, 139, 140]
|z〉 = ezĴ+
(1 + |z|2)j |j,−j〉, (8.3)
where |j,−j〉 is the state with the lowest pseudospin projection and z is a complex
number parameterized in terms of the two angles of the Bloch sphere as z = eiφ tan θ
2
(for details, see Appendix C). The function thus obtained is
Hcl(θ, φ) := 〈z|Ĥ|z〉 = −2hJz −
1
j
(J2
x + γJ2
y ), (8.4)
and it defines the classical energy surface where the dynamics of the pseudospin vector
J = j(sin θ cosφ, sin θ sinφ, cos θ) will take place. Explicitly, in terms of the angles (θ, φ),
the function
Hcl = −j[2h cos θ + sin2 θ(cos2 φ+ γ sin2 φ)] (8.5)
possesses two extrema, each of them defining a phase of the system. The two phases are:
• Symmetric phase: θ0 = 0. It corresponds to a classical pseudospin vector aligned
with the z axis. The ground state energy is Eg := Hcl(0, φ0) = −2hj.
• Broken phase: θ0 = cos−1 h with φ0 = 0 or φ0 = π. It corresponds to two
possible configurations of the pseudospin vector, signaling two ground states with
degenerate energy Eg := Hcl(cos
−1 h, 0) = Hcl(cos
−1 h, π) = −(1 + h2)j. In this
case, the classical pseudospin is not aligned with the z axis.
The ground state energy Eg = Eg(h) is thus the piecewise function
Eg
j
=
{
−(1 + h2), h < 1
−2h, h > 1
, (8.6)
which has a discontinuous second derivative at h = 1, signaling a second order quantum
phase transition [41]. The region h > 1 corresponds to the symmetric phase where
the ground state is unique, whereas the region h < 1 is the broken phase which has a
degenerate ground state energy.
Analysis in the thermodynamic limit 111
8.1.1 Symmetric phase
To carry out the analysis of the symmetric phase in the thermodynamic limit, we will
use the Holstein-Primakoff transformation [118]:
Ĵ+ =
√
2j
√
1− â†â
2j
â, Ĵ− =
√
2jâ†
√
1− â†â
2j
, Ĵz = j − â†â, (8.7)
which is then truncated to zeroth order in 1/j under the assumption that j → ∞. Hence,
we have
Ĵ+ ≃
√
2j â, Ĵ− ≃
√
2j â†, Ĵz = j − â†â. (8.8)
Taking this into account and using Ĵ± = Ĵx ± iĴy, the resulting quadratic Hamiltonian
that corresponds to (8.2) is
Ĥ ≃ −1 + γ
2
− 2hj − (1 + γ − 2h)â†â− 1− γ
2
(
â†2 + â2
)
, (8.9)
which in terms of Q̂ and P̂ can be written as
Ĥ ≃ −h− 2hj + (h− γ)P̂ 2 + (h− 1)Q̂2. (8.10)
From (8.9), it is clear that the Hamiltonian can be diagonalized through the Bogoliubov
transformation from operators (â, â†) to (b̂, b̂†) as
â = coshα b̂+ sinhα b̂†, ↠= sinhα b̂+ coshα b̂†, (8.11)
with tanh 2α = 1−γ
2h−γ−1 . By doing this, the Hamiltonian takes the form
Ĥ ≃ −h− 2hj + 2
√
(h− 1)(h− γ)
(
b̂†b̂+
1
2
)
. (8.12)
It is readily noted here that at the phase transition, h = 1, the frequency of the resulting
harmonic oscillator vanishes.
8.1.2 Broken phase
In the case of the broken phase, we need to perform a rotation around the y axis to align
the z axis with the pseudospin ground state configuration. Hence, we shall transform
the operators (Ĵx, Ĵy, Ĵz) to a new set of operators (Ĵ ′x, Ĵ
′
y, Ĵ
′
z) as




Ĵx
Ĵy
Ĵz




=




cos θ0 0 sin θ0
0 1 0
− sin θ0 0 cos θ0








Ĵ ′x
Ĵ ′y
Ĵ ′z




, (8.13)
112 Application to the anisotropic Lipkin-Meshkov-Glick model
where cos θ0 = h and sin θ0 =
√
1− h2, which is the ground state configuration that
corresponds to (θ0, φ0) = (cos−1 h, 0). Thus, the Hamiltonian of the broken phase turns
out to be [132, 133]
Ĥ ′ =− 2h2Ĵ ′z + 2h
√
1− h2Ĵ ′x −
1
j
(1− h2)Ĵ ′2z
− 1
j
[
h2Ĵ ′2x + h
√
1− h2(Ĵ ′xĴ
′
z + Ĵ ′zĴ
′
x) + γĴ ′2y
]
. (8.14)
Next, we apply the truncated Holstein-Primakoff transformation (8.8) to these rotated
operators to find the quadratic Hamiltonian for the broken phase. The resulting Hamil-
tonian is given by
Ĥ ′ ≃ −(1 + h2)j + (1− γ)P̂ 2 + (1− h2)Q̂2, (8.15)
or, in terms of the creation and annihilation operators b̂ and b̂†,
Ĥ ′ ≃ −(1 + h2)j + 2
√
(1− h2)(1− γ)
(
b̂†b̂+
1
2
)
. (8.16)
We observe once more that at the critical point, h = 1, the frequency of the resulting
harmonic oscillator vanishes, which signals the quantum phase transition. Now that
we have at hand the effective quadratic Hamiltonians for both the symmetric and the
broken phase, we proceed to the calculation of the classical and quantum metric tensors.
8.2 Classical and quantum metric tensors
We are now ready to compute the classical and quantum metrics in the thermodynamic
limit j → ∞. In what follows, we take x = {xi} = (h, γ), i = 1, 2 to be the adiabatic
parameters. To build the classical metric, we nee to derive the deformation functions
from the Hamiltonian (8.4). They are
O1 =
∂Hcl
∂h
= −2Jz,
O2 =
∂Hcl
∂γ
= −J
2
y
j
. (8.17)
At this point, we introduce canonical coordinates for the description of the classical
system. It is easy to see that the coordinates (φ, Jz) are canonical in the sense that they
reproduce the angular momentum algebra {Ji, Jj} = ǫijkJk, where
{f, g} =
∂f
∂φ
∂g
∂Jz
− ∂f
∂Jz
∂g
∂φ
. (8.18)
Then, we perform a canonical transformation and move to the (Q,P ) representation,
where
Q =
√
2(j − Jz) cosφ, P =
√
2(j − Jz) sinφ. (8.19)
Classical and quantum metric tensors 113
After this, the resulting classical LMG Hamiltonian is
Hcl = −2hj + h(P 2 +Q2)− (γP 2 +Q2)
(
1− P 2 +Q2
4j
)
. (8.20)
In Figure 8.1, we can see the level curves of the classical Hamiltonian Hcl in terms of
the (Q,P ) coordinates for the two different phases of the model. Once the mean field
Hamiltonian Hcl is constructed with the coherent states, the analysis is purely classical
in terms of fixed points and their stability. This highlights the importance of the classical
methods for quantum systems.
-1.5-1.0-0.5 0.0 0.5 1.0 1.5
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
Q
P
Hcl
-2.04
-1.36
-0.68
0
0.68
1.36
2.04
2.72
(a)
-1.5-1.0-0.5 0.0 0.5 1.0 1.5
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
Q
P
Hcl
-0.92
-0.69
-0.46
-0.23
0
0.23
0.46
0.69
(b)
Figure 8.1: Phase space corresponding to the classical Hamiltonian Hcl for γ = 0.1 and (a) h = 1.3,
or (b) h = 0.3. In (a), the red point is the only minimum; this is the symmetric phase. On the other
hand, in (b), the red point becomes a local maximum and the green points appear as two degenerate
minima; this is the broken phase.
8.2.1 Metrics for the symmetric phase
We first consider the symmetric phase. The quadratic Hamiltonian associated with
(8.20) is
Hcl ≃ −2hj + (h− γ)P 2 + (h− 1)Q2. (8.21)
Notice that when Q = P = 0, only the ground state energy of the symmetric phase,
Eg = −2hj, survives. We need to express the deformation functions (8.17) in terms of
initial action-angle variables and time. To do this, we find the solution to the equations
of motion of (8.21), which are
Q(t) =Q0 cosωt+
P0
ω
sinωt,
P (t) =P0 cosωt− ωQ0 sinωt, (8.22)
114 Application to the anisotropic Lipkin-Meshkov-Glick model
where we have identified the frequency as ω =
√
(h− 1)(h− γ). From here, we readily
find the action-angle variables (φ0, I) and write the initial conditions in terms of them
as
Q0 =
√
2I
√
h− γ√
h− 1
sinφ0, P0 =
√
2I
√
h− 1√
h− γ
cosφ0. (8.23)
We substitute (8.23) in (8.22) and use (4.75) to find the classical metric. The resulting
metric components are
g11 =
I2
32
[
1− γ
(h− 1)(h− γ)
]2
,
g12 =
I2(1− γ)
32(h− 1)(h− γ)2
,
g22 =
I2
32(h− γ)2
. (8.24)
We see that at the quantum phase transition, h = 1, the components g11 and g12 diverge.
Besides, one can easily note that the determinant of the classical metric is zero.
Now, to compute the quantum metric in the thermodynamic limit, we use the Hamil-
tonian (8.2) which leads to the deformation operators
Ô1 =
∂Ĥ
∂h
= −2Ĵz,
Ô2 =
∂Ĥ
∂γ
= − Ĵ
2
y
j
. (8.25)
Recall that when j → ∞, the truncated Holstein-Primakoff transformation allows us
to cast the angular momentum operators in the (Q̂, P̂ ) representation, as suggested by
equation (8.8). Thus, the deformation operators read
Ô1 = P̂ 2 + Q̂2 − 2j − 1,
Ô2 = −P̂ 2. (8.26)
We express them in terms of creation and annihilation operators and time, and read off
the spectrum from the effective quadratic Hamiltonian (8.10). This information is then
substituted into equation (2.56), which yields the following metric components:
g
(0)
11 =
1
32
[
1− γ
(h− 1)(h− γ)
]2
,
g
(0)
12 =
1− γ
32(h− 1)(h− γ)2
,
g
(0)
22 =
1
32(h− γ)2
. (8.27)
Classical and quantum metric tensors 115
This quantum metric has already been obtained in reference [17] by using another
method, hence, we corroborate it with our approach. We also see that the metric de-
terminant is zero, which was noted in the same reference and implies that information
geometry is ill-defined in the symmetric phase of this model. Comparing equations (8.24)
and (8.27), it is easy to see that the classical and quantum results have exactly the same
parameter dependence and that the classical metric reproduces the singularities of the
quantum metric. Moreover, both metrics match perfectly if the identification I2 = 1
is made. This is a remarkable result, because the classical metric is easier to compute:
it only requires the parameterization of the canonical coordinates (Q,P ) in terms of
action-angle variables (see equation (8.23)), instead of the introduction of creation and
annihilation operators. Furthermore, it uses the classical average instead of expectation
values of operators.
8.2.2 Metrics for the broken phase
For the broken phase, the classical Hamiltonian is obtained by taking the expectation
value of equation (8.14) in spin coherent states |z〉, which leads to
H ′cl =− j(1 + h2) + (1− γ)P 2 + (1− h2)Q2
+
h√
j
√
1− h2Q(P 2 +Q2)
√
1− P 2 +Q2
4j
+
1
4j
(P 2 +Q2)
[
γP 2 + h2Q2 − (1− h2)(P 2 +Q2)
]
. (8.28)
The quadratic approximation of this Hamiltonian is given by
H ′cl ≃ −j(1 + h2) + (1− γ)P 2 + (1− h2)Q2. (8.29)
In this case, the evaluation of H ′cl at Q = P = 0 yields the broken phase ground state
Eg = −j(1+h2). The deformation functions are the same as (8.17), however, they must
be expressed in terms of rotated quantities, in which case they take the following form:
O′1 = 2
√
1− h2J ′x − 2hJ ′z,
O′2 = −J
′2
y
j
. (8.30)
From here, we move to the position-momentum representation through the transforma-
tion (8.19) and find that the deformation functions are
O′1 = h(P 2 +Q2) + 2
√
j(1− h2)Q− 2hj − h,
O′2 = −P 2. (8.31)
116 Application to the anisotropic Lipkin-Meshkov-Glick model
We use (8.22) and (8.23), and substitute (8.31) into (4.75), which yields the metric
components
g11 =
jI
√
(1− h2)(1− γ)
+
I2
32
[
h(h2 − γ)
(1− h2)(1− γ)
]2
,
g12 =
I2h (h2 − γ)
32 (1− h2) (1− γ)2
,
g22 =
I2
32(1− γ)2
. (8.32)
As for the quantum metric in the broken phase, we use the deformation operators (8.25),
rewrite them in terms of the rotated angular momenta (8.13), and use the truncated
Holstein-Primakoff transformation (8.8) to compute the metric. The resulting quantum
metric tensor is
g
(0)
11 =
j
2
√
(1− h2)(1− γ)
+
1
32
[
h(h2 − γ)
(1− h2)(1− γ)
]2
,
g
(0)
12 =
h(h2 − γ)
32(1− h2)(1− γ)2
,
g
(0)
22 =
1
32(1− γ)2
. (8.33)
Again, it is remarkable that both the classical and quantum metrics have the same
parameter structure and match perfectly with the identifications I = 1/2 and I2 =
1. In contrast to the symmetric phase, both metrics are now invertible and have the
determinants
det gij =
I3
32
√
(1− h2)(1− γ)5
,
det g
(0)
ij =
j
64
√
(1− h2)(1− γ)5
, (8.34)
which, at the critical point, h = 1, diverge. This is a new result, since reference [17] did
not analyze the broken phase of the model. Thus, we have found that the broken phase
has a well-defined metric structure that allows a further geometric characterization with
the aid of the scalar curvature. In addition to this, we see once more that the classical
metric provides the same information as its the quantum counterpart but with a simpler
method. The scalar curvature for either of these two metrics can be computed with
equation (1.20), giving as a result
R = −4 +
7h4 − (9γ − 2)h2 − 4(1− γ)
j
√
(1− h2)(1− γ)3
. (8.35)
Classical and quantum metric tensors 117
From this expression, we observe that for large values of j (as expected in the thermo-
dynamic limit), the scalar curvature practically takes on the constant value −4, and it
diverges at h = 1, which indicates the presence of the quantum phase transition. It is
interesting to observe that the singularity of the metric is independent of the coordinate
system for this phase since it also appears in the scalar curvature. To investigate more
details of this phenomenon, we consider in the next section a numerical analysis for finite
j.
8.2.3 Finite-size analysis
We now want to address the effects of having a finite j directly and without resorting
to any approximations. The Hamiltonian (8.2) can be numerically diagonalized, so
we can compare the classical metric (equations (8.24) and (8.32)) with the quantum
metric (equations (8.27) and (8.33)) for a given value of j and see how well the analytic
computation via the truncated Holstein-Primakoff transformation agrees with the exact
quantum metric tensor1. We obtain the numerical results by employing the so called
perturbative form (2.64), which reads
g
(0)
ij (x) =
∑
n 6=0
〈0|Ôi|n〉〈n|Ôj|0〉
(En − E0)2
. (8.36)
The evaluation of this formula requires the time-independent deformation operators,
as opposed to equation (2.56), but at the cost of summing over all the elements of
the eigenspace of Ĥ. The Wolfram Mathematica code that was used to implement
equation (8.36) is shown in Appendix D. In Figure 8.2, we show the three components
of the quantum metric tensor for j = 100 and the scalar curvature obtained through
numerical differentiation over a mesh in the parameter space (see Appendix D). We see
the appearance of peaks near h = 1, which we identify as the precursors of the quantum
phase transition. This is most clearly seen in Figure 8.3, where we show the quantum
metric and its scalar curvature for a fixed γ and different values of j. We notice that
the peaks of the metric components and the scalar curvature become narrower and get
closer to h = 1 as j increases, which corroborates their identification as the precursors
of the quantum phase transition.
In Figure 8.4, we compare the classical metric in the thermodynamic limit with the
exact quantum metric tensor for j = 500 and γ = −0.5. We see that the agreement
between them is acceptable as long as we are not close to the phase transition, where
the Holstein-Primakoff approximation fails [128]. At the transition, the analytic metric
components g11 and g12 show a divergence that is not present in their finite j counterparts
coming from equation (8.36). Nevertheless, they still bear a resemblance given that, as
we mentioned, the exact metric components have peaks that grow and get closer to
1In this context, by “exact” we mean that we can reach sufficiently high digit accuracy with the use
of numerical diagonalization.
118 Application to the anisotropic Lipkin-Meshkov-Glick model
Figure 8.2: Metric components and scalar curvature for j = 100. The plots clearly show the presence
of the precursors of the quantum phase transition. The critical line h = 1 is shown in cyan.
h = 1 as j increases. Actually, this suggests that for large values of j and h = 1, the
components g11 and g12 will evolve to a divergent behavior. On the other hand, g22 does
not seem to have such a good agreement with its analytic counterpart near the phase
transition; this is because g22 (see equations (8.24) and (8.32)) is not sensitive to the
critical value h = 1, unlike the other components. For the scalar curvature, we see in
Figure 8.4 (d), that the analytic plot has a divergence at h = 1, and that it does not exist
in the region h > 1, which was expected from equation (8.35). However, the numerical
result indicates that the scalar curvature is smooth after h = 1 and Figure 8.3 shows
that the slope of the descending curve for h > 1 gets steeper as j increases. Thus, for
large values of j and h = 1, the scalar curvature will be an almost vertical line that falls
off to large negative values. This dissimilar behavior between the analytic and numerical
results is a consequence of the failure of the Holstein-Primakoff truncation at h = 1.
To better understand the behavior of the numerical quantum metric and its scalar
curvature, we close this section by presenting a fundamental result of our work: an
analysis of the quantum metric and its scalar curvature near the phase transition for
finite j. In Figure 8.5, we plot the height of the peaks of the metric components as a
Classical and quantum metric tensors 119
j=50
j=100
j=250
j=500
0.8 0.9 1.0 1.1 1.2
h0
500
1000
1500
g11
(a)
j=50
j=100
j=250
j=500
0.8 0.9 1.0 1.1 1.2
h
-6
-5
-4
-3
-2
-1
0
1
g12
(b)
j=50
j=100
j=250
j=500
0.8 0.9 1.0 1.1 1.2
h0.00
0.01
0.02
0.03
0.04
g22
(c)
j=50
j=100
j=250
j=500
0.7 0.8 0.9 1.0
h
-20
-15
-10
-5
0
R
(d)
Figure 8.3: Quantum metric tensor and its scalar curvature for different values of j when γ = −0.5.
The peaks in the components g11 and g12 become narrower as j increases.
function of h while we fix γ = −0.52. The curves that interpolate the points have the
following form:
• g11 peak
g
(peak)
11 = −22.5317 +
1.5333
(h− 1)2
. (8.37)
• g12 peaks
g
(peak 1)
12 = 0.0608 +
0.1990
h− 1
,
g
(peak 2)
12 = −0.0103− 0.0048
h− 1
. (8.38)
• g22 peaks
g
(peak 1)
22 = 0.0498− 0.0142h+ 0.0042h2,
g
(peak 2)
22 = 0.0046− 0.0042h+ 0.0019h2,
g
(peak 3)
22 = 0.0207− 0.0125h− 0.0194h2. (8.39)
2The peaks are numbered from left to right (see Figures 8.3 and 8.4).
120 Application to the anisotropic Lipkin-Meshkov-Glick model
0.7 0.8 0.9 1.0 1.1 1.2
h0
500
1000
1500
g11
(a)
0.8 0.9 1.0 1.1 1.2
h
-6
-4
-2
0
2
g12
(b)
0.7 0.8 0.9 1.0 1.1 1.2
h0.00
0.01
0.02
0.03
0.04
g22
(c)
0.7 0.8 0.9 1.0
h
-20
-15
-10
-5
0
5
R
(d)
Figure 8.4: Metric components and scalar curvature of the classical (or quantum) metric in the
thermodynamic limit (solid blue) and the exact quantum metric for j = 500 (dashed black) as a
function of h when γ = −0.5. The g11 and g12 components of the thermodynamic limit diverge at the
phase transition (dotted orange), while the quantum metric is always smooth.
From the functions (8.37) and (8.38), it is clear that in the limit h = 1, the metric
components g11 and g12 exhibit a divergent behavior, which is in accordance with the
corresponding classical (or quantum) metric components in the thermodynamic limit.
In Figure 8.6, we plot the height of the peaks as a function of j for γ = −0.5. A linear
relation between these quantities is evident when using a log-log scale. The function we
use to fit the points is
ln(g
(peak)
ij ) = m ln(j) + n, (8.40)
where the parameters m and n are shown in Table 8.1 for every peak. We reproduce the
value m ≈ 1.3 for the metric component g11 that was obtained in references [123, 132,
133]. In addition to this, we analyze the other components, finding that the sum of the
m values for the two peaks of g12 is 1.3142, which is a similar result to that of the g11
component. This is because g12 has mixed information about the parameters h and γ.
Accordingly, the m values of g22 do not relate with those of the other components.
Now, we study the scalar curvature for two representative values of γ. This will
allow us to characterize R for finite j and infer its behavior in the j → ∞ limit. We first
analyze the behavior of the extrema as functions of h. In Figure 8.7, we plot the two
Classical and quantum metric tensors 121
Peak m n
g
(peak)
11 1.3103 -0.7480
g
(peak 1)
12 0.6549 -2.2513
g
(peak 2)
12 0.6593 -4.1853
g
(peak 1)
22 -0.0081 -3.1676
g
(peak 2)
22 -0.0037 -6.0269
g
(peak 3)
22 0.0731 -4.9007
Table 8.1: Values of m and n in equation (8.40) for γ = −0.5. Notice that the two peaks of g12 have
the property that their values of m sum up to 1.3142, which is close to the m value of g11 peak.
peaks of R for γ = −0.5 and γ = −0.1. One of the peaks is a local minimum (peak 1)
and the other is a maximum (peak 2). The functions that interpolate the points of the
minima are
R(peak 1)(h, γ = −0.5) = −3.407− 2.197
h+ 0.795
,
R(peak 1)(h, γ = −0.1) = −3.443− 1.525
h+ 0.285
, (8.41)
whereas for the maxima they are
R(peak 2)(h, γ = −0.5) = −2.972 +
3.674
h+ 0.412
,
R(peak 2)(h, γ = −0.1) = −2.685 +
2.145
h− 0.073
. (8.42)
At h = 1, where the quantum phase transition appears in the thermodynamic limit, the
first peak of R takes the value of −4.631 when γ = −0.5 and −4.630 when γ = −0.1,
whereas the second peak goes to −0.370 when γ = −0.5 and to −0.371 when γ = −0.1.
The value of the first peak in both cases is very close to the one predicted by the truncated
Holstein-Primakoff approximation. Also, notice that at h = 1, the minima for the two
different values of γ are similar, which is also the case for the maxima. This is seen in
Figure 8.7, where both lines intersect.
If we now analyze the value of the peaks as a function of j, we can fit the data for
the minima as follows:
R(peak 1)(j, γ = −0.5) = −4.645− 3.882
j0.812
,
R(peak 1)(j, γ = −0.1) = −4.655− 6.100
j0.879
. (8.43)
122 Application to the anisotropic Lipkin-Meshkov-Glick model
For the maxima, the functions take the form
R(peak 2)(j, γ = −0.5) = −0.365 +
3.408
j0.695
,
R(peak 2)(j, γ = −0.1) = −0.360 +
4.749
j0.726
. (8.44)
We show in Figure 8.8 the behavior of the two peaks of R as a function of j when
γ = −0.5 and γ = −0.1. Two main features are seen. First, the minimum (peak 1)
grows as j increases and the maximum (peak 2) decreases with j. Second, the functions
(8.43) reveal that the minimum reaches a value around −4.6 when j → ∞, whereas the
maximum (8.44) goes to −0.36. This is also confirmed by equations (8.41) and (8.42),
since the limits h = 1 and j → ∞ predict the same results. The persistence of both
peaks when j → ∞ seems to indicate that the scalar curvature is not singular in the
thermodynamic limit; perhaps a more detailed analysis for higher values of j can help
understand this issue. Nevertheless, it is important to remark that despite the scalar
curvature’s smoothness for finite j, its peaks function as precursors of the quantum phase
transition.
A comment can be made regarding the change of sign in the maximum of R. Solving
the condition R(peak 2) = 0 yields the value of j for which the maximum is zero. When
γ = −0.5, this value is j = 25, while for γ = −0.1, the result is j = 35. This indicates
a local change in the geometry of the parameter space, i.e., a change in curvature from
spherical-type to hyperbolic-type, although we do not attribute any special interpretation
to those values of j.
We close by remarking that in reference [123], the authors found the fidelity suscep-
tibility related to the parameter h, however, we contributed with the exploration of the
parameter γ through the other two components of the quantum metric tensor. In this
way, we closed the geometric study with the analytic and numerical computation of the
scalar curvature, as well as with the analysis of their peaks.
Classical and quantum metric tensors 123
j=32
j=64
j=128
j=256
j=512
j=1024
j=2048
0.85 0.90 0.95 1.00
h0
2000
4000
6000
8000
10000
g11
(peak)
(a)
j=32
j=64
j=128
j=256
j=512
j=1024
j=2048
0.85 0.90 0.95 1.00
h
-15
-10
-5
0
g12
(peak 1)
(b)
j=32
j=64
j=128
j=256
j=512
j=1024
j=2048
0.96 0.97 0.98 0.99 1.00
h0.0
0.5
1.0
1.5
2.0
2.5
g12
(peak 2)
(c)
j=32
j=64
j=128
j=256
j=512
j=1024
0.80 0.85 0.90 0.95
h
0.0400
0.0405
0.0410
0.0415
g22
(peak 1)
(d)
j=32
j=64
j=128
j=256
j=512
j=1024
0.92 0.94 0.96 0.98
h
0.00236
0.00237
0.00238
0.00239
0.00240
g22
(peak 2)
(e)
j=32
j=64
j=128
j=256
j=512
j=1024
1.06 1.08 1.10 1.12 1.14 1.16
h0.0090
0.0095
0.0100
0.0105
0.0110
0.0115
0.0120
0.0125
g22
(peak 3)
(f)
Figure 8.5: Peaks of the components of the quantum metric as functions of h when γ = −0.5. The
value of j is indicated for each point. The g11 component has only one peak, g12 has two peaks, and
g22 has three peaks.
124 Application to the anisotropic Lipkin-Meshkov-Glick model
4 5 6 7
Ln( j)
-5
5
10
Ln(|gij
(peaks)
|)
Ln(g11
(peak)) Ln(g12
(peak 1))
Ln(g12
(peak 2)) Ln(g22
(peak 1))
Ln(g22
(peak 2)) Ln(g22
(peak 3))
Figure 8.6: Peaks of the components of the quantum metric as functions of j when γ = −0.5. The
axes are presented in logarithmic scale.
γ=-0.5γ=-0.1
0.5 0.6 0.7 0.8 0.9 1.0
h
-5.3
-5.2
-5.1
-5.0
-4.9
-4.8
-4.7
R(peak 1)
(a)
γ=-0.5γ=-0.1
0.75 0.80 0.85 0.90 0.95 1.00
h
-0.2
0.0
0.2
0.4
R(peak 2)
(b)
Figure 8.7: Peaks of the scalar curvature as functions of h for γ = −0.5 and γ = −0.1. The values of
j=12,16,20,24,28,32,40,50,75,100,125,175,250,300,500 were considered. In both plots, j grows with h.
Classical and quantum metric tensors 125
γ=-0.5γ=-0.1
100 200 300 400 500
j
-5.3
-5.2
-5.1
-5.0
-4.9
-4.8
-4.7
R(peak 1)
(a)
γ=-0.5γ=-0.1
100 200 300 400 500
j
-0.2
0.0
0.2
0.4
R(peak 2)
(b)
Figure 8.8: Peaks of the scalar curvature as functions of j for γ = −0.5 and γ = −0.1. The minimum
is shown in (a) and the maximum is shown in (b).

Chapter 9
Application to a modified
Lipkin-Meshkov-Glick model
In this chapter, we present a Lipkin-Meshkov-Glick (LMG) model with a linear term in
Ĵx. In this case, the parameter space provides an invertible metric in both phases, and
therefore, a well-defined scalar curvature across the parameter space. We first build the
classical Hamiltonian using Bloch coherent states and find its fixed points. We will focus
on the quantum phase transition that appears in the highest energy state, so we com-
pute the quantum geometric tensor for this state using the truncated Holstein-Primakoff
transformation. As a result, we observe a divergence in the metric components at the
phase transition and we find that the Berry curvature only exists in one of the phases.
We then compute the scalar curvature and show that it has a singularity at the phase
transition. We then contrast the analytic results with their finite-size counterparts ob-
tained through exact numeric diagonalization, and find a good agreement between them,
except in points near the phase transition where the Holstein-Primakoff approximation
ceases to be valid. We finally include a detailed analysis of the precursors of the quantum
phase transition that appear in the metric and its scalar curvature.
The modified LMG model that we propose has the Hamiltonian
ĤLMG = ΩĴz + ΩxĴx +
ξy
j
Ĵ2
y , (9.1)
where Ĵx,y,z = (1/2)
∑
i σ
(i)
x,y,z are the collective pseudo-spin operators and j = N /2, with
j coming from the eigenvalue j(j + 1) of the total spin operator Ĵ2 = Ĵ2
x + Ĵ2
y + Ĵ2
z .
Also, Ω, Ωx and ξy are real parameters, and we put ~ = 1. We begin by studying the
properties of this model when j → ∞, i.e., the thermodynamic limit.
9.1 Analysis in the thermodynamic limit
The classical LMG Hamiltonian, which gives an exact description of the quantum system
for j → ∞, is obtained by taking the expectation value of ĤLMG/j in Bloch coherent
127
128 Application to a modified Lipkin-Meshkov-Glick model
states |z〉 =
(
1 + |z|2
)−j
ezĴ+ |j,−j〉, where |j,−j〉 is the state with the lowest pseudo-
spin projection, and Ĵ+ is the raising operator (for details, see Appendix C). Defining
z = e−iφ tan θ
2
, the classical LMG Hamiltonian has the simple form
Hcl =
〈z|ĤLMG|z〉
j
= −Ωcos θ + Ωx sin θ cosφ+ ξy sin
2 θ sin2 φ, (9.2)
or, in terms of the canonical variables
Q =
√
2(1− cos θ) cosφ, P = −
√
2(1− cos θ) sinφ, (9.3)
it reads
Hcl(Q,P ) =
Ω
2
(
P 2 +Q2
)
− Ω + ΩxQ
√
1− P 2 +Q2
4
+ ξyP
2
(
1− P 2 +Q2
4
)
. (9.4)
We now set out to find the fixed points of the Hamiltonian (9.4). To simplify the
analysis, we consider only two independent parameters (Ωx and ξy), setting from now
on Ω = 1. Notice that the Hamiltonian (9.2) is invariant under φ → −φ, which has as
a consequence the invariance of (9.4) under P → −P . This implies that all the fixed
points are doubly degenerate if φ 6= 0 or φ = π.
The fixed points of (9.4) are found by imposing the conditions Q̇ = 0 and Ṗ = 0 on
the Hamilton equations in the region 0 ≤ θ ≤ π, 0 ≤ φ < 2π, which means that
Q̇ =
∂Hcl
∂P
=
P
2
[
2Ω− ξy(2P
2 +Q2 − 4)− ΩxQ
√
4− (P 2 +Q2)
]
= 0,
Ṗ = −∂Hcl
∂Q
=
1
2
[
ξyP
2Q+
ΩxQ
2
√
4− (P 2 +Q2)
− Ωx
√
4− (P 2 +Q2)− 2ΩQ
]
= 0, (9.5)
The valid solutions of this system are:
1. For any value of Ωx and ξy:
• x1 = (Q1, P1) =
(
√
2 + 2√
Ω2
x+1
, 0
)
with (θ1, φ1) =
(
arccos
(
− 1√
Ω2
x+1
)
, 0
)
.
• x2 = (Q2, P2) =
(
−
√
2− 2√
Ω2
x+1
, 0
)
with (θ2, φ2) =
(
arccos
(
1√
Ω2
x+1
)
, π
)
.
2. For ξy ≤ −
√
Ω2
x+1
2
:
• x3 = (Q3, P3) =
(
− Ωx√
ξy(2ξy−1)
,−
√
4ξ2y−Ω2
x−1
ξy(2ξy−1)
)
with (θ3, φ3) =
(
arccos
(
− 1
2ξy
)
, arccos
(
− Ωx√
4ξ2y−1
))
and sinφ3 =
√
1− cos2 φ3.
Analysis in the thermodynamic limit 129
• x
′
3 = (Q′3, P
′
3) =
(
− Ωx√
ξy(2ξy−1)
,
√
4ξ2y−Ω2
x−1
ξy(2ξy−1)
)
with (θ′3, φ
′
3) =
(
arccos
(
− 1
2ξy
)
, arccos
(
− Ωx√
4ξ2y−1
))
and sinφ′3 = −
√
1− cos2 φ′3.
3. For ξy ≥
√
Ω2
x+1
2
:
• x4 = (Q4, P4) =
(
Ωx√
ξy(2ξy−1)
,−
√
4ξ2y−Ω2
x−1
ξy(2ξy−1)
)
with (θ4, φ4) =
(
arccos
(
− 1
2ξy
)
, arccos
(
Ωx√
4ξ2y−1
))
and sinφ4 =
√
1− cos2 φ4.
• x
′
4 = (Q′4, P
′
4) =
(
Ωx√
ξy(2ξy−1)
,
√
4ξ2y−Ω2
x−1
ξy(2ξy−1)
)
with (θ′4, φ
′
4) =
(
arccos
(
− 1
2ξy
)
, arccos
(
Ωx√
4ξ2y−1
))
and sinφ′4 = −
√
1− cos2 φ′4.
The energies hi = Hcl(xi) associated with each fixed point are given in Table 9.1.
Point Energy
x1
√
Ω2
x + 1
x2 −
√
Ω2
x + 1
x3 and x
′
3
(4ξ2y + Ω2
x + 1)/(4ξy)
x4 and x
′
4
(4ξ2y + Ω2
x + 1)/(4ξy)
Table 9.1: Classical energies at each fixed point.
Some properties of the energies are:
• The energies h1 and h2 are defined for any value of Ωx and ξy.
• The energy h3 is only defined in the region ξy ≤ −
√
Ω2
x+1
2
, where it satisfies h3 ≤ h2.
• The energy h4 is only defined in the region ξy ≥
√
Ω2
x+1
2
, where it satisfies h4 ≥ h1.
Each of the curves ξy = ±
√
Ω2
x+1
2
is the separatrix between different regions which cor-
respond to different quantum phases of the LMG model in the thermodynamic limit.
Accordingly, for a fixed ξy, the values Ωx = ±Ωxc = ±
√
4ξ2y − 1 represent the critical
coupling at which the quantum phase transition occurs. To understand this, we show in
Figure 9.1 the energies as functions of Ωx for a fixed ξy. In (a), for negative ξy, we clearly
see that the ground state energy has the value h3 when −Ωxc < Ωx < Ωxc, while it takes
130 Application to a modified Lipkin-Meshkov-Glick model
the value h2 when Ωx < −Ωxc or Ωx > Ωxc; this indicates the presence of a quantum
phase transition in the ground state. However, for positive ξy, the ground state energy
is given only by h2 in the whole region, as clearly shown in (b), thus, no phase transition
appears in this case.
Regarding the highest energy, for negative ξy, we see in (a) that it takes the value
h1 for any Ωx so it does not present a phase transition. On the other hand, when ξy is
positive, (b) shows that the highest energy is h4 in the region −Ωxc < Ωx < Ωxc, whereas
for Ωx < −Ωxc or Ωx > Ωxc, it takes the value h1; therefore, we have a quantum phase
transition in this case.
In what follows, we only consider positive values of ξy so that we have the quantum
phase transition occurring in the highest energy state. Due to the symmetries of the
model, reversing the sign of ξy would generate an analogous description for the ground
state. For ξy <
√
Ω2
x+1
2
, the highest energy state is unique and we call it the symmetric
phase. On the contrary, when ξy >
√
Ω2
x+1
2
, the highest energy state is doubly degenerate
and we call it the broken phase.
-6 -4 -2 2 4 6
Ωx
-6
-4
-2
2
4
6
h
h1 h2 h3
-6 -4 -2 2 4 6
Ωx
-6
-4
-2
2
4
6
h
h1 h2 h4
(a) (b)
Figure 9.1: Classical energies as functions of Ωx when (a) ξy = −2.3 and (b) ξy = 2.3. The energies
h3 and h4 are only defined in the region −4.490 ≤ Ωx ≤ 4.490 whose extreme values are marked with
the dashed gray lines.
We show in Figure 9.2 the expectation values of some observables in Bloch coher-
ent states and their comparison with the expectation values taken in the highest energy
eigenstate. The quantum phase transition manifests as a discontinuity in the first deriva-
tive of the plots at Ωxc = 4.490 for (b), (c), and (d), whereas as a discontinuity in the
second derivative at Ωxc = 4.490 for (a). The coherent states give an excellent description
of the system even for j = 128.
To further illustrate the quantum phase transition, we show in Figure 9.3 the fixed
points x1, x2, x4, and x
′
4. We can see how the fixed points move and how the phase
space trajectories change around them when the parameter Ωx is varied leaving ξy fixed.
Analysis in the thermodynamic limit 131
-5 5
Ωx
2
4
6
8
〈H LMG〉/j
-5 5
Ωx
-1.0
-0.5
0.5
1.0
〈Jx〉/j
(a) (b)
-5 5
Ωx0.14
0.16
0.18
0.20
0.22
〈Jz〉/j
-5 5
Ωx
0.2
0.4
0.6
0.8
1.0
〈Jy2〉/ j2
(c) (d)
Figure 9.2: Comparison of the expectation values of different observables taken in the highest energy
eigenstate (dashed blue) and in spin coherent states (solid orange) for j = 128 and ξy = 2.3. The critical
values ±Ωxc = ±4.490 are signaled with the gray dashed lines.
The green point is x2 and it corresponds to the ground state of the system; as we know,
for positive values of ξy the ground state does not present a quantum phase transition,
thus preserving a center stability. The two blue points are x4 and x
′
4, and for Ωx < Ωxc,
they correspond to the highest energy states; they are stable center points in this region.
As we increase Ωx, we see that the two blue points become closer until they disappear
at the phase transition when Ωx = Ωxc. Finally, the red point is x1 and it corresponds
to the highest energy state when Ωx > Ωxc, where it is stable center point. However,
note that at Ωx = Ωxc it changes its stability, becoming an unstable hyperbolic point for
Ωx < Ωxc and having a positive Lyapunov exponent given by
λ =
√
√
1 + Ω2
x (2ξy −
√
1 + Ω2
x). (9.6)
Figure 9.4 shows the Lyapunov exponent of x1 as a function of the parameters Ωx and
ξy. It is important to mention that the Hamiltonian (9.4) is integrable and thus has
regular dynamics. The Lyapunov exponent is therefore not associated to chaos but to
132 Application to a modified Lipkin-Meshkov-Glick model
an instability.
In the quantum domain, the instability of x1 is associated with an excited-state
quantum phase transition (ESQPT). A main signature of ESQPTs is the divergence of
the density of states at an energy denoted by Ec. In the mean-field approximation, it
has been shown that this energy coincides with the energy of the classical system at the
saddle point [141, 142], in this case, Ec = jHcl(x1) = jh1 = j
√
Ω2
x + 1. In Figure 9.5, we
show the density of states for the LMG model taking j = 256, ξy = 2 and Ωx = 0.774.
These parameters correspond to the unstable region where there is a non-zero Lyapunov
exponent (see Figure 9.4). The peak in the density of states associated with the ESQPT
is clearly visible at h1, marked with a vertical red dashed line.
Analysis in the thermodynamic limit 133
(a) Ωx = 0.2Ωxc (b) Ωx = 0.6Ωxc
(c) Ωx = Ωxc (d) Ωx = 2Ωxc
Figure 9.3: Level curves of the classical LMG Hamiltonian Hcl(Q,P ) = E for different values of Ωx
and fixing ξy = 2. The green point is x2 and is stable; the blue points are x4 and x
′
4 and are also stable;
the red point is stable when Ωx > Ωxc (d), but unstable when Ωx < Ωxc ((a) and (b)), where it has a
positive Lyapunov exponent.
134 Application to a modified Lipkin-Meshkov-Glick model
Figure 9.4: Lyapunov exponent for the critical point x1 as a function of the coupling parameters Ωx
and ξy. The black zone indicates a null Lyapunov exponent.
-1.0 -0.5 0.0 0.5 1.0 1.5 2.0
0
5
10
15
E/j
N
u
m
b
er
o
f
st
at
es
Figure 9.5: Density of states when ξy = 2 and Ωx = 0.2Ωxc for j = 256. The red dashed line indicates
the classical energy h1 =
√
Ω2
x + 1 = 1.265 where the ESQPT takes place.
Quantum geometric tensor for the LMG model 135
9.2 Quantum geometric tensor for the LMG model
In this section, we study the geometry of the ground state and the highest energy states
of the LMG model. While the semiclassical analysis performed above employing SU(2)
coherent states provides rich and valuable information about the system, it is not well
suited to study the geometry of the parameter space of this system. As shown in Ap-
pendix E, the metric obtained employing coherent states is ill-defined for ξy <
√
Ω2
x+1
2
since it has a vanishing determinant. To overcome this difficulty, we first compute the
quantum metric tensor and its scalar curvature in the thermodynamic limit via the
truncated Holstein-Primakoff transformation [118]. This approximation becomes exact
in this limit as long as we are not close to a phase transition, where spurious divergences
may appear [128]. After the analytic computation, we carry out an exact diagonalization
and compare the analytic and numeric results.
In what follows, we take x = {xi} = (Ωx, ξy) with i = 1, 2 as the adiabatic parameters.
As we said before, we only focus on positive values of ξy, so that the phase transition
occurs for the highest energy state.
9.2.1 Ground state
We begin our analysis with the ground state, which corresponds via the classical Hamil-
tonian (9.4) to the fixed point x2 (the green one in Figure 9.3) with angular coordinates
(θ2, φ2) =
(
arccos
(
1√
Ω2
x+1
)
, π
)
. The first step is to align the classical pseudospin of
the ground state with the z axis. To do this, we perform a rotation of the spin operators
around the y axis as follows [132, 133]




Ĵx
Ĵy
Ĵz




=




cos θ2 0 sin θ2
0 1 0
− sin θ2 0 cos θ2








Ĵ ′x
Ĵ ′y
Ĵ ′z




. (9.7)
With this rotation, the Hamiltonian (9.1) takes the form
Ĥ =
√
1 + Ω2
xĴ
′
z +
ξy
j
Ĵ ′2y , (9.8)
which is suitable for applying the Holstein-Primakoff transformation that maps angular
momentum operators into bosonic operators as
Ĵ ′z = â†â− j, Ĵ ′+ =
√
2j â†
√
1− â†â
2j
, Ĵ ′− =
√
2j
√
1− â†â
2j
â. (9.9)
It is readily verified that this representation satisfies the SU(2) algebra as long as [â, â†] =
1. Now, we consider the thermodynamic limit j → ∞ and expand the square roots
136 Application to a modified Lipkin-Meshkov-Glick model
retaining only the zeroth order term in 1/j [123, 132, 133]. The Cartesian components
of the angular momentum turn out to be
Ĵ ′x ≃
√
j
2
(
↠+ â
)
, Ĵ ′y ≃ −i
√
j
2
(
↠− â
)
, Ĵ ′z = â†â− j. (9.10)
In principle, we may substitute these expressions into equation (9.8) and perform a
Bogoliubov transformation to creation and annihilation operators (b̂, b̂†) that diagonalize
the Hamiltonian. However, for illustrative purposes, we make first the intermediate
transformation Q̂ = 1√
2
(
↠+ â
)
, P̂ = i√
2
(
↠− â
)
to find that the Hamiltonian takes the
form
Ĥ ≃ −j
√
Ω2
x + 1 +
(
√
Ω2
x + 1 + 2ξy
2
)
P̂ 2 +
√
Ω2
x + 1
2
Q̂2. (9.11)
This suggests the use of the following transformation
Q̂ =
(
√
1 + Ω2
x + 2ξy
4
√
1 + Ω2
x
)1/4
(
b̂† + b̂
)
, P̂ = i


√
1 + Ω2
x
4
(
√
1 + Ω2
x + 2ξy
)


1/4
(
b̂† − b̂
)
,
(9.12)
that will cast the Hamiltonian (9.11) into the harmonic oscillator
Ĥ ≃ −j
√
Ω2
x + 1 +
√
√
Ω2
x + 1
(
√
Ω2
x + 1 + 2ξy
)
(
b̂†b̂+
1
2
)
, (9.13)
which has the frequency ω =
√
√
Ω2
x + 1
(
√
Ω2
x + 1 + 2ξy
)
and energy E = −j
√
Ω2
x + 1.
The quantum metric tensor can now be calculated with the aid of equation (2.64) setting
n = 0 and employing the operators
∂ĤLMG
∂Ωx
=Ĵx =
1
√
Ω2
x + 1
Ĵ ′x +
Ωx
√
Ω2
x + 1
Ĵ ′z,
∂ĤLMG
∂ξy
=
Ĵ2
y
j
=
Ĵ ′2y
j
, (9.14)
provided that they are expressed in terms of (b̂, b̂†) to act on the eigenstates of Ĥ. After
doing this, we find the components
g11 =
j
2 (Ω2
x + 1)7/4
√
√
Ω2
x + 1 + 2ξy
+
ξ2yΩ
2
x
8 (Ω2
x + 1)2
(
√
Ω2
x + 1 + 2ξy
)2 ,
g12 = − ξyΩx
8 (Ω2
x + 1)
(
√
Ω2
x + 1 + 2ξy
)2 ,
g22 =
1
8
(
√
Ω2
x + 1 + 2ξy
)2 , (9.15)
Quantum geometric tensor for the LMG model 137
and the determinant
g =
j
16 (Ω2
x + 1)7/4
(
√
Ω2
x + 1 + 2ξy
)5/2
. (9.16)
We see that a singularity appears in all the components when ξy = −
√
Ω2
x+1
2
. Also, we
observe that g11 consists of two terms, one of them is proportional to j and dominant
as j → ∞. Retaining all the terms and using equation (1.20), we find that the scalar
curvature associated of the metric (9.15) simplifies to
R = −4. (9.17)
Remarkably, the scalar curvature is constant despite the non-trivial dependence of the
metric components on the parameters. Furthermore, there is no sign of the singularity
that appeared in the metric, at least in the region of the parameter space under con-
sideration. We must mention that if we had considered only the dominant term in j
in g11 for the computation of the scalar curvature, the result would have been different;
therefore, we choose to retain all the terms. This constant negative curvature signals
that the parameter space of the ground state possesses a hyperbolic geometry and is
isomorphic to the Lobachevsky space [51].
In Figure 9.6, we plot the metric components and the scalar curvature for ξy = 2.3
and j = 120. We see that the numeric results agree well with their analytic counterparts
coming from the Holstein-Primakoff approximation. Figure 9.7 shows a map of the
scalar curvature for different values of j, where we see a tendency to the analytic result
as j increases. Notice that the singularity predicted by the metric components (9.15) is
outside the range of the parameters employed in the plots (ξy > 0).
138 Application to a modified Lipkin-Meshkov-Glick model
Analytic
Numeric
1 2 3 4 5 6
Ωx0
5
10
15
20
25
g11
Analytic
Numeric
1 2 3 4 5 6
Ωx
-0.004
-0.003
-0.002
-0.001
0.000
0.001
0.002
0.003
g12
(a) (b)
Analytic
Numeric
1 2 3 4 5 6
Ωx0.000
0.001
0.002
0.003
0.004
g22
Analytic
Numeric
1 2 3 4 5 6
Ωx
-4.00
-3.98
-3.96
-3.94
-3.92
-3.90
R
(c) (d)
Figure 9.6: Metric components and scalar curvature for the ground state when j = 120 and ξy = 2.3.
Quantum geometric tensor for the LMG model 139
(a) j = 20 (b) j = 50
(c) j = 80 (d) j = 120
Figure 9.7: Scalar curvature map of the ground state for different values of j.
140 Application to a modified Lipkin-Meshkov-Glick model
9.2.2 Highest energy state
We now study the highest energy state for which the quantum phase transition occurs.
We divide the parameter space into regions above and below the separatrix, and thus,
treat each phase separately. The energy of the maximum is (see Figure 9.1)
Eh
j
=
{
4ξ2y+Ω2
x+1
4ξy
, Ωx < Ωxc
√
Ω2
x + 1 , Ωx > Ωxc
, (9.18)
The energy Eh and its first derivative are continuous as a function of Ωx at the critical
point Ωx = Ωxc, whereas its second derivative presents a discontinuity. This signals a
second order quantum phase transition.
9.2.2.1 Symmetric phase
We begin with the region in parameter space for ξy <
√
Ω2
x+1
2
(symmetric phase). Here,
the procedure to find the quantum metric tensor is similar to that of the ground state.
The angular coordinates of the corresponding fixed point x1 (the red one in Figure
9.3) are (θ1, φ1) =
(
arccos
(
− 1√
1+Ω2
x
)
, 0
)
, and the rotation that aligns the classical
pseudospin with the z axis is




Ĵx
Ĵy
Ĵz




=




cos θ1 0 − sin θ1
0 1 0
sin θ1 0 cos θ1








Ĵ ′x
Ĵ ′y
Ĵ ′z




. (9.19)
Making use of the truncated Holstein-Primakoff transformation, we arrive at the quadratic
Hamiltonian
Ĥ ≃ j
√
Ω2
x + 1−
(
√
Ω2
x + 1− 2ξy
2
)
P̂ 2 −
√
Ω2
x + 1
2
Q̂2, (9.20)
which is a harmonic oscillator with frequency ω =
√
√
Ω2
x + 1
(
√
Ω2
x + 1− 2ξy
)
and
energy E = j
√
Ω2
x + 1. Notice that there is a minus sign in the last two terms of equation
(9.20) due to this fixed point being a maximum of energy, although the dynamics is still
Quantum geometric tensor for the LMG model 141
oscillatory. The components of the metric are
g11 =
j
2 (Ω2
x + 1)7/4
√
√
Ω2
x + 1− 2ξy
+
ξ2yΩ
2
x
8 (Ω2
x + 1)2
(
√
Ω2
x + 1− 2ξy
)2 ,
g12 = − ξyΩx
8 (Ω2
x + 1)
(
√
Ω2
x + 1− 2ξy
)2 ,
g22 =
1
8
(
√
Ω2
x + 1− 2ξy
)2 , (9.21)
and the determinant is
g =
j
16 (Ω2
x + 1)7/4
(
√
Ω2
x + 1− 2ξy
)5/2
. (9.22)
We can easily identify a singularity in all the components of the metric and the determi-
nant which occurs at ξy =
√
Ω2
x+1
2
. This is precisely the critical point where the quantum
phase transition takes place, and confirms the usefulness of the quantum metric tensor
to detect a quantum phase transition. With the quantum metric tensor (9.21) at hand,
we compute its scalar curvature with the aid of (1.20) and find
R = −4, (9.23)
which again means that the underlying geometry is hyperbolic.
9.2.2.2 Broken phase
Now, when ξy >
√
Ω2
x+1
2
(broken phase), there are two fixed points that correspond to
the highest energy (the blue ones in Figure 9.3). Choosing x4, which has the angular
coordinates (θ4, φ4) =
(
arccos
(
− 1
2ξy
)
, arccos
(
Ωx√
4ξ2y−1
))
, the rotation that aligns the
classical pseudospin with the z axis is




Ĵx
Ĵy
Ĵz




=




cosφ4 − sinφ4 0
sinφ4 cosφ4 0
0 0 1








cos θ4 0 − sin θ4
0 1 0
sin θ4 0 cos θ4








Ĵ ′x
Ĵ ′y
Ĵ ′z




. (9.24)
Following similar steps as with the normal phase, we find that the quadratic Hamiltonian
now is
Ĥ ≃j 4ξ
2
y + Ω2
x + 1
4ξy
− ξy
(
4ξ2y − Ω2
x − 1
)
4ξ2y − 1
P̂ 2 +
Ωx
√
4ξ2y − Ω2
x − 1
2
(
4ξ2y − 1
)
(
Q̂P̂ + P̂ Q̂
)
− 16ξ4y − 8ξ2y + Ω2
x + 1
4ξy
(
4ξ2y − 1
) Q̂2. (9.25)
142 Application to a modified Lipkin-Meshkov-Glick model
This Hamiltonian has the form of a generalized harmonic oscillator. In order to remove
the crossed term in Q̂ and P̂ , we make the further linear transformation
Q̂ =
√
2ξy
(
4ξ2y − Ω2
x − 1
)
4ξ2y − 1
Q̂′, P̂ =
√
4ξ2y − 1
2ξy
(
4ξ2y − Ω2
x − 1
) P̂ ′ +
Ωx
√
2ξy
(
4ξ2y − 1
)
Q̂′.
(9.26)
The Hamiltonian then turns into a simple harmonic oscillator
Ĥ = j
4ξ2y + Ω2
x + 1
4ξy
− P̂ ′2
2
− ω2
2
Q̂′2, (9.27)
with frequency ω =
√
4ξ2y − Ω2
x − 1 and energy E = j
4ξ2y+Ω2
x+1
4ξy
. Again, two minus signs
appear since we are considering a maximum of energy, although the dynamics is still
oscillatory.
The resulting expressions for the metric components in this case are cumbersome,
however, all of them are singular at points on the separatrix. Also, the computation of
the scalar curvature via equation (1.20) yields a cumbersome expression. In Figure 9.8,
we show the quantum metric tensor and its scalar curvature as well as their numeric
counterparts, observing a good agreement between them, except for g11 near Ωx = 0.
This component has a maximum at Ωx = 0 which is not observed in the result of the
analytic Holstein-Primakoff calculation and it seems to be related to the vanishing of 〈Ĵx〉
when Ωx changes sign as observed in Figure 9.2 (b). In any case, this strong difference
does not affect much the scalar curvature, which approaches to zero in the broken phase
in both the analytic and numeric results. We must say, as explained in detail in Appendix
F, that the numerical evaluation of the metric demands extremely high precision (from
15 to 60 decimal digits), involving extensive computational resources.
Now, it turns that under the analytic computation the broken phase exhibits a Berry
curvature, whose only component is given by
F12 = − 2j + 1
4ξ2y
√
4ξ2y − Ω2
x − 1
+
16ξ2y − Ω2
x + 1
16ξ3y
(
4ξ2y − Ω2
x − 1
) . (9.28)
We observe, as expected, that the Berry curvature is also singular at the separatrix. On
the other hand, the numeric analysis yields a zero Berry curvature. The reason for this
discrepancy is that the Berry phase appeared due to the rotation that was performed to
align the highest energy state with the classical pseudospin, introducing in this way the
crossed term in Q̂ and P̂ in the Hamiltonian (9.25).
In Figure 9.9, the plots of the metric components and the scalar curvature for the
highest energy state are shown in 3D, while Figure 9.10 contains their maps. We see
that in the symmetric phase, the scalar curvature has a value around -4, just as the
Holstein-Primakoff calculation predicts. However, as Ωx decreases, the scalar curvature
begins growing, passes the separatrix, takes small positive values, reaches a maximum
and then descends to near zero values.
Quantum geometric tensor for the LMG model 143
0 1 2 3 4 5 6
Ωx0
2000
4000
6000
8000
g11
0 1 2 3 4 5 6
Ωx0.1
1
10
100
1000
104
g11
Numeric Analytic
Numeric
Analytic
1 2 3 4 5 6
Ωx
-30
-25
-20
-15
-10
-5
0
g12
(a) g11 (b) g12
Numeric
Analytic
0 1 2 3 4 5 6
Ωx0
10
20
30
40
50
60
g22
Numeric
Analytic
1 2 3 4 5 6
Ωx
-4
-3
-2
-1
0
R
(c) g22 (d) R
Figure 9.8: Comparison of the metric components and the scalar curvature with the analytic results
for ξy = 2.3 and j = 96. The inset shows the g11 component in logarithmic scale. The agreement is
excellent except near to the quantum phase transition (dashed gray). Notice the difference in g11between
the analytic and numeric curves as Ωx → 0. Also, the analytic plots show a divergence at Ωxc due to
the unreliability of the truncated Holstein-Primakoff approximation at that point.
Now a question arises. How can we interpret the change of sign near the separatrix?
Does it mean that there is a change in the topology of the parameter space and that it
switches between a closed and an open shape? To answer this, we must recall that a
fundamental quantity in the study of two-dimensional surfaces is the Gaussian curvature
K, which is related to the scalar curvature as K = R/2. It is defined as the product of
the two principal curvatures, κ1 and κ2, which quantify the bending of the surface along
each direction [143]. If both principal curvatures, κ1 and κ2, have the same sign, then
the Gaussian curvature is positive and the local geometry is spherical. On the contrary,
when both principal curvatures have opposite signs, the Gaussian curvature is negative
and the surface is locally hyperbolic. A change in topology would imply that the metric
becomes singular at some point [144], which means that its determinant should vanish
there. Since we do not observe that the metric determinant is zero (see Figure 9.9 (e)),
we conclude that a change in topology does not take place. Instead, there is only a sign
change in one of the two principal curvatures, κ1 or κ2, producing a local change between
dome-like and saddle-like shapes.
In Figure 9.11, we show the plots of the quantum metric tensor and its scalar cur-
vature for various values of j when ξy = 2.3. We can see how the peaks of the metric
144 Application to a modified Lipkin-Meshkov-Glick model
components get sharper and closer to the critical value Ωxc = 4.490 as j increases, thus
anticipating that in the limit j → ∞ they will diverge (they are the precursors of the
quantum phase transition for finite j). Also, the maximum of the scalar curvature gets
closer to the separatrix, and the transition region between the asymptotic values −4 on
one side and 0 on the other becomes thinner, suggesting that a discontinuity will appear
at Ωxc in the thermodynamic limit. In the next section, we carry out a detailed analysis
to extract more relevant information about this behavior.
Quantum geometric tensor for the LMG model 145
(a) g11 (b) g11 in log scale
(c) g12 (d) g22
(e) g in log scale (f) R
Figure 9.9: Metric components, determinant, and scalar curvature for the highest energy state with
j = 32. The cyan line is the separatrix ξy =
√
1+Ω2
x
2 .
146 Application to a modified Lipkin-Meshkov-Glick model
(a) g11 (b) g12
(c) g22 (d) R
Figure 9.10: Maps of the metric components and the scalar curvature for the highest energy state
with j = 32.
Quantum geometric tensor for the LMG model 147
1 2 3 4 5 6
Ωx0
5000
10000
15000
g11
0 1 2 3 4 5 6
Ωx0.1
1
10
100
1000
10
4
g11
1 2 3 4 5 6
Ωx
-30
-25
-20
-15
-10
-5
0
g12
(a) g11 (b) g12
0 1 2 3 4 5 6
Ωx0
10
20
30
40
50
60
g22
1 2 3 4 5 6
Ωx
-4
-3
-2
-1
0
R
(c) g22 (d) R
Figure 9.11: Metric components and scalar curvature for the highest energy state with ξy = 2.3 and
j = 32, 48, 64, 96, 128. The inset shows the g11 component in logarithmic scale.
148 Application to a modified Lipkin-Meshkov-Glick model
9.2.3 Analysis of the peaks
We now analyze the peaks of the metric components and the scalar curvature to infer
their behavior in the thermodynamic limit. We consider three interesting cases: (i) fixing
ξy = 2.3, (ii) fixing Ωx =0, and (iii) fixing ξy = 0.5.
9.2.3.1 Case 1: ξy = 2.3
In Figure 9.12, we plot the peak (maximum or minimum) of every metric component
and of the scalar curvature as a function of Ωx for ξy = 2.3 and increasing values of j.
The functions used to fit the data are:
g
(max)
11 = 0.582 +
2.051
(Ωx − 4.490)2
,
g
(min)
12 = 2.249− 2.966
(Ωx − 4.490)2
,
g
(max)
22 = −7.170 +
5.931
(Ωx − 4.490)2
,
R(max) = −0.083 + 2.470 e−0.356Ωx . (9.29)
We see that at the critical point, Ωxc = 4.490, the metric components are singular,
just as the analytic formulas (9.21) predict. On the other hand, the maximum of the
scalar curvature takes the value 0.416. This is an indication that the maximum of the
scalar curvature persists and that there is not a singularity when j → ∞, but rather a
discontinuity produced by a sudden change of sign across the quantum phase transition,
which is implied by the increasing slope of R seen in Figure 9.11 (d).
Next, in Figure 9.13, we show the peaks of the metric components and the scalar
curvature as functions of j. In this case, the functions used to fit the data are:
log(g
(max)
11 ) = −3.102 + 1.267 log(j),
log(g
(min)
12 ) = −3.134 + 1.349 log(j),
log(g
(max)
22 ) = −2.702 + 1.394 log(j),
R(max) = 0.418 +
1.563
(j − 0.913)0.680
. (9.30)
In the thermodynamic limit j → ∞, the metric components diverge, whereas the scalar
curvature approaches 0.418, which agrees with the prediction of the previous analysis
when Ωx = Ωxc. Once again, this confirms that R is not singular across the quantum
phase transition and that it just presents a sudden sign change, i.e., a discontinuity. As
a consequence, it can be said that the singularity that appears in the metric is removable
and is not a true singularity of the parameter space.
Quantum geometric tensor for the LMG model 149
3.7 3.8 3.9 4.0 4.1 4.2 4.3
Ωx
20
40
60
80
100
120
g11
max
3.8 3.9 4.0 4.1 4.2 4.3
Ωx
-200
-150
-100
-50
g12
min
(a) (b)
3.8 3.9 4.0 4.1 4.2 4.3
Ωx
100
200
300
400
g22
max
3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5
Ωx0.40
0.45
0.50
0.55
Rmax
(c) (d)
Figure 9.12: Behavior of the maximum of each metric component and the scalar curvature with
respect to Ωx for ξy = 2.3. The points correspond to j = 32, 48, 64, 96, 128, 160, 192, 256, 300, 384, 512.
9.2.3.2 Case 2: Ωx = 0
Now, we analyze the value of the metric components g11 and g22, and the scalar curvature
when Ωx = 0. The plots are shown in Figure 9.14 and the functions that fit the data
are:
g011 = (−0.013 + 0.976j)2,
g022 = 0.005 + 0.005j,
R0 =
1
0.131 + 0.238j
. (9.31)
Notice that g12 does not appear because its value at Ωx = 0 is zero. Remarkably, the
scalar curvature goes to zero as j → ∞, which is precisely the prediction of the coherent-
state approach shown in Appendix E (see equation E.6).
9.2.3.3 Case 3: ξy = 0.5
Finally, we show in Figure 9.15 the peaks of the metric components g11 and g22, as
well as the scalar curvature for ξy = 0.5. This case is particularly interesting, since
150 Application to a modified Lipkin-Meshkov-Glick model
log(g11
max )
log(|g12
min |)
log(g22
max )
3.5 4.0 4.5 5.0 5.5 6.0
log(j)
2
3
4
5
6
100 200 300 400 500
j
0.44
0.46
0.48
0.50
0.52
0.54
0.56
0.58
Rmax
(a) Maxima of |gij |. (b) R(max).
Figure 9.13: Maximum of each metric component and the scalar curvature with respect to j for
ξy = 2.3.
at ξy = 0.5 the peaks of the quantum metric tensor and the scalar curvature occur at
Ωx = 0 regardless of the value of j. The functions used to fit the data in Figure 9.15 are:
log(g011) = −0.480 + 1.325 log(j),
log(g022) = −1.881 + 1.327 log(j),
R0 = −2.183 +
3.430
(j2 − 6.206)0.284
. (9.32)
In this case, in the thermodynamic limit j → ∞, the metric components also diverge,
whereas the scalar curvature approaches −2.183. This is a limiting point, over the
separatrix, where R takes an intermediate value between the asymptotic ones in the two
phases (0 and -4). Once again, this confirms that R is not singular at the quantum phase
transition.
In summary, in this chapter we obtained a geometrical characterization of the mod-
ified LMG model with the aid of the quantum metric tensor and its scalar curvature.
In particular, our analysis for finite j allows us to deduce that the second-order quan-
tum phase transition in this model is indicated by the sudden sign chane in the scalar
curvature. Thus, the singularity predicted by the analytic scalar curvature at Ωxc is a
consequence of the unreliability of the truncated Holstein-Primakoff approximation in a
close vicinity of the phase transition. In this sense, one may try to use the resummation
of the Holstein-Primakoff series proposed in [145] and see if the results obtained with
this technique coincide with the numerical predictions.
Quantum geometric tensor for the LMG model 151
100 200 300 400 500
j
100
200
300
400
500
g11
0
50 100 150 200 250
j
0.2
0.4
0.6
0.8
1.0
1.2
g22
0
(a) (b)
50 100 150 200 250
j
10
20
30
40
50
60
1/R0
(c)
Figure 9.14: Plots of g11, g22, and R with respect to j for ξy = 2.3 and Ωx = 0.
log(g11
0 )
log(g22
0 )
3.5 4.0 4.5 5.0 5.5 6.0
log(j)
3
4
5
6
7
8 100 200 300 400 500
j
-2.2
-2.1
-2.0
-1.9
-1.8
-1.7
R0
(a) Maxima of gij . (b) R0.
Figure 9.15: Maximum of each metric component and the scalar curvature with respect to j for
ξy = 0.5.

Chapter 10
Relativistic Coulomb problem from
N = 4 super Yang-Mills
In this chapter, we analyze how to generalize the Runge-Lenz (RL) vector to the rela-
tivistic Coulomb problem. In Section 5.4, we discussed the two-dimensional attractive
Coulomb problem in the non-relativistic setting. Although we did not use it, the RL
vector can be introduced to find straightforwardly the equation of trajectory. In the rel-
ativistic case, the minimal coupling prescription on the Klein-Gordon equation destroys
the SO(4) symmetry and hence no RL vector exists. Thus, our interest is the modi-
fication of the coupling prescription to allow for a RL vector that restores the SO(4)
symmetry.
We see in this chapter that N = 4 super Yang-Mills (SYM) gives rise to a modified
Klein-Gordon equation by means of a specific Higgs mechanism. In particular, we present
the simplest spontaneous breaking of symmetry that allows one to extract this integrable
system where the Coulomb potential is coupled to both the mass and the energy of the
particle. Then, we see the deduction of this equation from a modified action principle for
a relativistic particle. Next, we obtain the RL vector and show that it generates, along
with the angular momentum, the SO(4) algebra, providing the relativistic spectrum
of the modified Klein-Gordon equation. Our results confirm previous analyses using
different approaches [146–148]. We finally make use of the Kustaanheimo-Stiefel (KS)
transformation and show that the relativistic spectrum of the hydrogen-like atom in two
dimensions is related to the relativistic harmonic oscillator spectrum in two dimensions.
Again, we confirm the analysis where the spectrum of the relativistic harmonic oscillator
was constructed by directly solving the modified Klein-Gordon equation [149]. The
results of this chapter have been published in references [150] and [151].
10.1 Modified Klein-Gordon equation from N = 4 SYM
Our interest in this section is to make more transparent the extraction of the modi-
fied Klein-Gordon equation (Klein-Gordon equation with scalar and vector potentials of
153
154 Relativistic Coulomb problem from N = 4 super Yang-Mills
equal magnitude, which is useful to describe the pseudospin symmetry in atomic nuclei
[152]) from N = 4 SYM. The usual Klein-Gordon equation describing a charged spinless
particle in a Coulomb field is obtained by the minimal coupling prescription ∂µ → Dµ =
∂µ +
i
~c
Aµ, where a particular reference frame is chosen to have Aµ = (−α/r,0),
−
(
1
c
∂
∂t
− i
~c
α
r
)2
φ+∇2φ−
(mc
~
)2
φ = 0. (10.1)
Expanding, we get
− 1
c2
∂2φ
∂t2
+∇2φ+
2iα
~c2r
∂φ
∂t
+
α2
~2c2r2
φ−
(mc
~
)2
φ = 0. (10.2)
Now, the modified Klein-Gordon equation includes a non-minimal coupling through the
mass,
−
(
1
c
∂
∂t
− i
~c
α
r
)2
φ+∇2φ−
(mc
~
− α
~cr
)2
φ = 0. (10.3)
Expanding this equation, it can be seen that the modification of the mass results in the
cancellation of the quadratic potential term,
− 1
c2
∂2φ
∂t2
+∇2φ+
2iα
~c2r
∂φ
∂t
+
2αm
~2r
φ−
(mc
~
)2
φ = 0. (10.4)
For future treatment, we write the relativistic form of this equation,
∂µ∂
µφ− 2iAµ∂
µφ− AµA
µφ−
(
m− α
r
)2
φ = 0, (10.5)
where ~ = c = 1 and the metric is ηµν = diag(−1, 1, 1, 1).
To undertake the spontaneous breaking of symmetry that will allow us to extract
the above equation, we consider only the bosonic sector of N = 4 SYM which has the
following Lagrangian density [153]
L = Tr
{
−1
4
F µνFµν −
1
2
6
∑
i=1
DµΦiD
µΦi +
g2
4
6
∑
i,j=1
[Φi,Φj]
2
}
. (10.6)
Here, the six scalar fields are N × N traceless Hermitian matrices in the adjoint repre-
sentation of SU(N). The action of the covariant derivative on a generic field W is given
by
DµW = ∂µW − ig[Aµ,W ], (10.7)
where under a gauge transformation U , the gauge field Aµ and the scalar fields Φi
transform as
Φi → UΦiU
†, Aµ → UAµU
† − i
g
(∂µU)U
†, (10.8)
Modified Klein-Gordon equation from N = 4 SYM 155
and as usual, the field strength Fµν is defined by the commutator of the covariant deriva-
tives
Fµν =
i
g
[Dµ, Dν ] = ∂µAν − ∂νAµ − ig[Aµ, Aν ]. (10.9)
The resulting equations of motion are [154]
DµF
νµ = ig
6
∑
i=1
[Φi, D
νΦi], (10.10)
DµD
µΦi = g2
6
∑
j=1
[Φj,[Φj,Φi]]. (10.11)
We choose to work with the group SU(2) for simplicity. Therefore, the fields will be
expressible in terms of the Pauli matrices τa as
Φi = Φa
i
τa
2
=
1
2
(
Φ0
i Φ−i
Φ+
i −Φ0
i
)
, Aµ = Aaµ
τa
2
=
1
2
(
A3
µ A1
µ − iA2
µ
A1
µ + iA2
µ −A3
µ
)
, (10.12)
where Φ±i = Φ1
i ± iΦ2
i . We introduce the Higgs mechanism by giving a vacuum expecta-
tion value v to Φ1 [148, 155, 156]
Φ1 =
1
2
(
Φ0
i + v 0
0 −Φ0
i − v
)
, (10.13)
and taking the other fields as
Φ2 =
1
2
(
0 Φ−2
Φ+
2 0
)
, Φi = 0, i = 3, 4, 5, 6; Aµ = Aaµ
τa
2
=
1
2
(
A3
µ 0
0 −A3
µ
)
. (10.14)
Now, the Lagrangian (10.6) reads
L =− 1
8
F µνFµν −
1
4
∂µΦ
−
2 ∂
µΦ+
2 − 1
4
∂µΦ
0
1∂
µΦ0
1 +
ig
4
Aµ(Φ−2 ∂µΦ
+
2 − Φ+
2 ∂µΦ
−
2 )
− g2
4
Φ−2 Φ
+
2 AµA
µ − g2
4
(Φ0
1 + v)2Φ−2 Φ
+
2 . (10.15)
Here, we have defined Aµ = A3
µ and Fµν = ∂µAν − ∂νAµ. A crucial step is to implement
the constraint
Φ0
1 +
α
r
= 0 (10.16)
in the previous Lagrangian. After the strong implementation of the constraint, we ob-
tain1
L =− 1
8
F µνFµν −
1
4
∂µΦ
−
2 ∂
µΦ+
2 +
ig
4
Aµ(Φ−2 ∂µΦ
+
2 − Φ+
2 ∂µΦ
−
2 )
− g2
4
Φ−2 Φ
+
2 AµA
µ − g2
4
(
v − α
r
)2
Φ−2 Φ
+
2 (10.17)
1This constraint can be interpreted as a second class constraint in Dirac’s sense.
156 Relativistic Coulomb problem from N = 4 super Yang-Mills
up to a boundary term. Thus, the field content of our theory has been reduced to one
vector field Aµ and one complex scalar Φ−2 . It is worth noting that this Lagrangian is
not Lorentz-invariant unless the coupling α is allowed to transform in such a way that
it cancels the contribution of 1/r.
The equation of motion for the scalar field Φ−2 arising from (10.17) is
∂µ∂
µΦ−2 − ig(∂µA
µ)Φ−2 − 2igAµ∂
µΦ−2 − g2AµA
µΦ−2 − g2(v + Φ0
1)
2Φ−2 = 0. (10.18)
Of course, the equation for Φ+
2 will be the complex conjugate of this one. Denoting
Φ−2 = φ and taking the Coulomb potential Aµ = (−α/r,0) with g = 1,2 this equation
becomes
∂µ∂
µφ− 2iAµ∂
µφ− AµA
µφ−
(
m− α
r
)2
φ = 0, (10.19)
which is clearly (10.5). We see that the vacuum expectation value of Φ1 is the mass m
of the scalar field φ, and that the imposed constraint provides the non-minimal coupling
m → m − α/r necessary to enhance the symmetry of the field theory from SO(3) to
SO(4). The Lagrangian (10.17) can be rewritten as
L =
1
2
[
−1
4
F µνFµν −
1
2
(Dµφ)
∗(Dµφ)− 1
2
(
m− α
r
)2
φ∗φ
]
, (10.20)
where the covariant derivative is given by Dµ = ∂µ − iAµ. We can recognize this as one
half the Lagrangian for scalar electrodynamics with modified mass. Now, we obtain the
equation of motion for the gauge field (10.10),
− ∂µ∂
µAν + ∂ν(∂ · A) = i
2
(φ ∂νφ∗ − φ∗∂νφ)− Aν |φ|2. (10.21)
The application of the divergence to (10.21) reveals that the conserved current is
Jν =
i
2
(φ ∂νφ∗ − φ∗∂νφ)− Aν |φ|2, (10.22)
which means that the density ρ is
ρ =
i
2
(φ∗∂tφ− φ ∂tφ
∗) +
α
r
|φ|2. (10.23)
On the other hand, if we set Aµ = (−α/r,0) in the equation of motion (10.21), we find
∇2
(α
r
)
=
i
2
(φ∗∂tφ− φ ∂tφ
∗) +
α
r
|φ|2, (10.24)
which implies
ρ = ∇2
(α
r
)
, (10.25)
just as expected, or, upon integration,
∫
d3r ρ = −4πα. (10.26)
2This is equivalent to absorbing the coupling constant into α.
Modified relativistic particle 157
10.2 Modified relativistic particle
In this section, we will see how the equation (10.19) possesses an enhanced symmetry
SO(4) that allows to write it in a Schrödinger-like form. We start by considering the
action for a relativistic particle with electromagnetic coupling
S =
∫
dτ L =
∫
dτ
(
−mc
√
−ηµν ẋµẋν +
1
c
Aµẋ
µ
)
. (10.27)
The momentum pµ conjugate to xµ is
pµ =
∂L
∂ẋµ
=
mcẋµ
√
−ηαβẋαẋβ
+
1
c
Aµ. (10.28)
Therefore, by considering the square of pµ − Aµ/c, we obtain the constraint
(
pµ −
1
c
Aµ
)(
pµ − 1
c
Aµ
)
+m2c2 = 0. (10.29)
Taking Aµ = (−α/r,0), the constraint reads
(p0)2 +
2α
cr
p0 +
α2
c2r2
− p2 −m2c2 = 0. (10.30)
If we apply the quantization prescription pµ → −i~∂µ and let this constraint act on a
scalar field φ, we obtain (10.1). Following the method of separation of variables, we write
the wave function as
φ(r, t) = exp
(
− i
~
Et
)
ϕ(r). (10.31)
This leads to the stationary Klein-Gordon equation with Coulomb potential that has the
well known spectrum [148, 157]
Enℓ = m

1 +
(
γ
n− (ℓ+ 1/2) +
√
(ℓ+ 1/2)2 − γ2
)2


−1/2
, n = 0, 1, 2, ... , (10.32)
where γ = α/~c. We can see the breaking of the n2 degeneracy of the hydrogen atom
due to the appearance of the orbital quantum number ℓ in the spectrum.
The classical stationary problem with conserved angular momentum L and energy
E = cp0 is given by the constraint (10.30)
− (E + α/r)2
c2
+ (p2r + L2/r2) +m2c2 = 0. (10.33)
The solutions of this equation for bounded orbits are rosettes [158], in contrast with the
non-relativistic problem where the orbits are ellipses [76]. As a consequence, the RL
158 Relativistic Coulomb problem from N = 4 super Yang-Mills
vector is not conserved. Now comes the question of finding the action that upon the
application of the previous method, will yield the modified Klein-Gordon equation that
possesses the SO(4) symmetry. It is clear from (10.3) that the mass should be affected
by the addition of the potential, so that now
S =
∫
dτ L =
∫
dτ
[
−mc
√
−
(
1− α
mc2r
)2
ηµν ẋµẋν +
1
c
Aµẋ
µ
]
. (10.34)
This action corresponds to a relativistic particle interacting with a Coulomb potential
in a curved spacetime described by the conformally flat metric
gµν =
(
1− α
mc2r
)2
ηµν . (10.35)
The momentum now becomes
pµ = mc
(
1− α
mc2r
) ẋµ
√
−ηαβẋαẋβ
+
1
c
Aµ, (10.36)
so that
(
pµ −
1
c
Aµ
)(
pµ − 1
c
Aµ
)
+m2c2
(
1− α
mc2r
)2
= 0. (10.37)
Again, taking Aµ = (−α/r,0), the constraint is reduced to
(p0)2 +
2α
cr
p0 +
2αm
r
− p2 −m2c2 = 0, (10.38)
which reproduces the modified Klein-Gordon equation (10.4) after quantization. Apply-
ing separation of variables in the resulting equation (10.4), we find
E2
~2c2
ϕ+∇2ϕ+
2αE
~2c2r
ϕ+
2αm
~2r
ϕ−
(mc
~
)2
ϕ = 0. (10.39)
This can be cast in the form of a Schrödinger-like equation
− ~
2
(E/c2 +m)
∇2ϕ− 2α
r
ϕ = (E −mc2)ϕ (10.40)
with the spectrum [148]
En = m
(
1− 2γ2
n2 + γ2
)
, n = 1, 2, 3, ... . (10.41)
It is noteworthy that we have recovered the degeneracy since the orbital quantum number
ℓ does not appear in this formula. This points to the existence of an additional integral
of motion, i.e., the relativistic generalization of the RL vector which will enable us to
Relativistic SO(4) algebra using the relativistic Runge-Lenz vector 159
recover the hidden SO(4) symmetry. In this case, the constraint (10.38) leads to the
classical problem
p2r + L2/r2
E/c2 +m
− 2α
r
= E −mc2. (10.42)
The quantum and classical equations suggest that we should make the identifications
E/c2 +m↔ 2ms, E −mc2 ↔ Es, 2α ↔ αs, (10.43)
to recover a Schrödinger equation with mass ms, energy Es, and coupling constant αs.
Notice that under the previous substitutions, the spectrum (10.41) can be easily obtained
from that of the hydrogen atom. In the next section, we will construct the relativistic
RL vector and obtain the energy levels à la Pauli [159].
10.3 Relativistic SO(4) algebra using the relativistic
Runge-Lenz vector
In this section, we obtain the relativistic RL vector associated with the modified rela-
tivistic Coulomb problem and we show that the orbit can be reconstructed using this
conserved quantity. We then proceed to the construction of the infinitesimal Noether
symmetries generated by the relativistic RL vector. We also describe the complete SO(4)
algebra generated by the angular momentum and the RL vector and recover the correct
relativistic spectrum of the corresponding hydrogen-like atom. We shall take units such
that ~ = c = 1.
Using as a model the non-relativistic construction [76] we find
d
dt
[
p×L− (E +m)
αr
r
]
= 0, (10.44)
where L is the angular momentum that generates the SO(3) algebra. This indicates that
the relativistic generalization of the RL vector is
A = p×L− (E +m)
αr
r
. (10.45)
The vector A enhances the symmetry from SO(3) to SO(4), and in this way we have
shown that the non-minimal coupling m→ m−α/r, or equivalently, the transformation
to a conformally flat space with metric (10.35) allows us to restore the SO(4) symmetry
in the relativistic case. Notice that by taking the non-relativistic limit of the RL vector,
we obtain
ANR = p×L− 2αmr
r
, (10.46)
which is the usual RL vector with a coupling constant twice as larger. We observe that
the analogous procedure to obtain the classical non relativistic orbit of the Coulomb
160 Relativistic Coulomb problem from N = 4 super Yang-Mills
problem can also be implemented in the relativistic case. If we take the dot product of
the RL vector with r, we get
A · r = Ar cos θ = (p×L) · r − (E +m)
αr · r
r
, (10.47)
or
1
r
=
α(E +m)
L2
[
1 +
A
α(E +m)
cos θ
]
. (10.48)
This is precisely the equation of an ellipse with one of its foci at the origin. The eccen-
tricity and semi-latus rectum are given, respectively by
e =
A
α (E +m)
, p =
L2
α(E +m)
, (10.49)
while the semi-major and semi-minor axes are
a =
αL2(E +m)
α2(E +m)2 − A2
, b =
L2
√
α2(E +m)2 − A2
. (10.50)
Now, we focus on the infinitesimal transformations associated with the position xi
and the canonical momentum pi that are generated by our relativistic RL vector. We
find that
δxi =
{
xi, ǫjAj
}
= 2(ǫ · r)pi − xi(ǫ · p)− (r · p)ǫi,
δpi =
{
pi, ǫjAj
}
= −p2ǫi + (p · ǫ)pi − (E +m)α
(
ǫ · r
r3
xi − ǫi
r
)
.
The RL vector also acts on the magnitude r as
δr =
{√
xlxl, ǫiAi
}
=
(p · r) (r · ǫ)
r
− (ǫ · p)r. (10.51)
These infinitesimal symmetries do not exactly correspond with the symmetry transfor-
mations previously written in [40]. A crucial difference is that in the approach given
in [40] the symmetry transformation acts in a dual momentum space (dual conformal
transformation) that is appropriate to reveal the symmetries of scattering amplitudes in
SYM theory.
We turn to the spectrum. It turns out that it can also be constructed from the
relativistic SO(4) algebra generalizing the non relativistic result as presented in [160].
We introduce a redefinition of the RL vector (10.45)
A′ =
2
E +m
A =
1
E +m
(p×L−L× p)− 2α
r
r
.
This vector A′ satisfies
[A′, H] = 0, L ·A′ = A′ ·L,
Relativistic SO(4) algebra using the relativistic Runge-Lenz vector 161
and
A′2 = 4
[
α2 +
E −m
E +m
(1 + L2)
]
. (10.52)
We can see that the corresponding relativistic algebra closes as
[Li, Lj] = iεijkLk,
[A′i, Lj] = iεijkA
′
k,
[A′i, A
′
j] = −4i
(
E −m
E +m
)
εijkLk.
Defining
D =
√
− E +m
4(E −m)
A′
and
M =
1
2
(L−D), N =
1
2
(L+D),
it is easy to show that the original algebra splits into the product of two SO(3) algebras
[Mi,Mj] = iεijkMk, [Ni, Nj] = iεijkNk,
with the constraint
M 2 = N 2. (10.53)
The operator M 2 +N 2 will have the eigenvalues 2ℓ(ℓ+1) with ℓ = 0, 1, 2, ... because of
the constraint (10.53). On the other hand, we can find that
M 2 +N 2 =
1
2
[
L2 − E +m
4(E −m)
A′2
]
= −1
2
(
E +m
E −m
α2 + 1
)
,
where we used (10.52). With this at hand, we can find the spectrum
En = m
(
1− 2α2
n2 + α2
)
. (10.54)
This is the same spectrum that we obtained through the identifications (10.43) and that
is reported in [148], reproduced here with n = 2ℓ + 1. Due to the hidden symmetry
lying under this non-minimal coupling which reveals the existence of the relativistic RL
vector, N = 4 SYM is dubbed as the “hydrogen atom quantum field theory” [161].
162 Relativistic Coulomb problem from N = 4 super Yang-Mills
10.4 Relativistic Kustaanheimo-Stiefel duality
In this section, we relate the wave functions and the spectra of the modified Coulomb
problem and the modified relativistic harmonic oscillator. We consider the illustrative
case of two dimensions (for the treatment of an arbitrary number of dimensions see
[150]). At the end, we see how the integrals of motion of both problems are related by
the KS duality.
The KS transformation relates the original variables (x, y, t) with new ones (u, v, s)
by [87]
x = u2 − v2, y = 2uv,
dt
ds
= u2 + v2 = r. (10.55)
We denote by φ̄(u, v, s) the field that results from evaluating the original field in terms
of the new variables,
φ̄(u, v, s) = φ(x(u, v), y(u, v), t(s)). (10.56)
Then, the transformation of the differential operators is
∂φ
∂t
=
∂φ̄
∂s
ds
dt
=
1
u2 + v2
∂φ̄
∂s
, (10.57)
∂2φ
∂t2
=
1
(u2 + v2)2
∂2φ̄
∂s2
, (10.58)
∇2φ =
1
4(u2 + v2)
∇2
uφ̄, (10.59)
where
∇2
uφ̄ =
∂2φ̄
∂u2
+
∂2φ̄
∂v2
. (10.60)
Thus, the modified Klein-Gordon equation (10.4) becomes
− 1
c2(u2 + v2)2
∂2φ̄
∂s2
+
1
4(u2 + v2)
∇2
uφ̄+
2iα
~c2(u2 + v2)
∂φ̄
∂s
+
2αm
~2(u2 + v2)
φ̄−
(mc
~
)2
φ̄ = 0. (10.61)
The separation (10.31) in terms of new variables is
φ̄(u, v, s) = exp

− i
~
E
s
∫
ds′ (u2 + v2)

 ϕ̄(u, v). (10.62)
Substituting it in (10.61) we get
− ~
2
4 (E/c2 +m)
∇2
uϕ̄− (E −mc2)(u2 + v2)ϕ̄ = 2αϕ̄. (10.63)
This is the stationary Schrödinger equation for a two-dimensional harmonic oscillator
with mass 2(E/c2 +m), energy 2α, and frequency
ω = c
√
−E −mc2
E +mc2
. (10.64)
Relativistic Kustaanheimo-Stiefel duality 163
Now, the time-dependent Schrödinger equation for the harmonic oscillator is
− ~
2
4M∇2
uξ̄ +
Mω2
2
(u2 + v2)ξ̄ = i~
∂ξ̄
∂s
, (10.65)
where
ξ̄(u, v, s) = exp
(
− i
~
Es
)
ϕ̄(u, v) = exp
(
− i
~
2αs
)
ϕ̄(u, v). (10.66)
Using (10.62) to substitute ϕ̄ in terms of φ̄ we get
ξ̄(u, v, s) = exp

− i
~

2αs− E
s
∫
ds′ (u2 + v2)



 φ̄(u, v, s)
= exp
[
− i
~
F (s)
]
φ̄(u, v, s). (10.67)
Here, F (s) is the generating function of the canonical (and non-holonomic) KS transfor-
mation on the extended phase space [162]. Thus, we have related the wave functions φ̄
of the Coulomb problem (written in terms of new variables) with the wave functions ξ̄ of
the harmonic oscillator. It is also seen that the wave function ϕ solves both stationary
equations. A final word of caution is that there will be sums in (10.67) over both E and
2α so the correspondence between the two wave functions is not one-to-one.
Now, we try to establish the relation between the spectra of the modified relativistic
harmonic oscillator and the modified Coulomb problem. In the Schrödinger case, the re-
lation between the harmonic oscillator mass, coupling constant, energy, and angular mo-
mentum (Ms, ks, Es,Ls) and the same variables of the Coulomb problem (ms, αs, Es, Ls)
is
Ms = 4ms, ks = −2Es, Es = αs, Ls = 2Ls. (10.68)
We know that the identification (10.43) allows us to map the relativistic problem onto
the non relativistic one, so this suggests the following steps to recover the spectrum of the
modified relativistic Coulomb problem from that of the modified relativistic harmonic
oscillator:
1. Make the identification (10.43) in the modified relativistic harmonic oscillator spec-
trum.
2. Apply the KS duality (10.68).
3. Make again the identification (10.43) in the opposite sense to obtain the modified
Coulomb problem spectrum.
We start with the spectrum of the modified relativistic harmonic oscillator in two di-
mensions [149]
(E/c2 +M)(E −Mc2)2
~2k
= (4q + 2|L|+ 2)2, q = 0, 1, 2, ... . (10.69)
164 Relativistic Coulomb problem from N = 4 super Yang-Mills
After carrying out step 1, we get the Schrödinger spectrum of the two-dimensional har-
monic oscillator
Es = ~
√
ks
Ms
(2q + |L|+ 1). (10.70)
Now, step 2 yields
Es = − msα
2
s
2~2(q + |L|+ 1/2)2
, (10.71)
which is the spectrum of the non relativistic two-dimensional hydrogen atom. Finally,
implementation of step 3 results in
E = mc2
[
1− 2γ2
(q + |L|+ 1/2)2 + γ2
]
, (10.72)
where γ = α/~c. By making n = q+ |L|, we readily recognize this as the spectrum of the
modified relativistic Coulomb problem in two dimensions. With these results at hand,
we are able to build the relativistic KS transformation duality (cf. (10.68)):
E/c2 +m =
1
4
(E/c2 +M),
E −mc2 = −k,
α =
1
2
(E −Mc2), (10.73)
which performs the mapping
P
2
E/c2 +M + k(u2 + v2) = E −Mc2 =⇒ p2
E/c2 +m
− 2α
r
= E −mc2. (10.74)
Let us now address the issue of how the integrals of motion of both problems are
related. We start with the non-relativistic case and then through the replacements
(10.43) we move to the modified relativistic problem. We have already seen that the
energy of a system becomes the coupling constant of the other one, so we analyze only
the remaining integrals of motion. For the two-dimensional Coulomb problem, they are
the Runge-Lenz vector As and the angular momentum L. The Runge-Lenz vector is
As x = xp2y − ypxpy −
msαsx
√
x2 + y2
,
As y = yp2x − xpxpy −
msαsy
√
x2 + y2
, (10.75)
and the angular momentum reads Ls = xpy − ypx. On the other hand, for the two-
dimensional harmonic oscillator we can find a conserved tensor Cs ij whose components
are [76]
Cs 11 =
P2
u
2Ms
+
k
2
u2,
Relativistic Kustaanheimo-Stiefel duality 165
Cs 12 =
PuPv
2Ms
+
k
2
uv,
Cs 22 =
P2
v
2Ms
+
k
2
v2, (10.76)
and the angular momentum Ls = upv − vpu. A straightforward calculation employing
the non-relativistic KS duality (10.68) shows that the mapping is
Cs 11 =
P2
u
2Ms
+
k
2
u2 = −As x
2ms
+
αs
2
,
Cs 12 =
PuPv
2Ms
+
k
2
uv = −As y
2ms
,
Cs 22 =
P2
v
2Ms
+
k
2
v2 =
As x
2ms
+
αs
2
,
Ls = 2Ls. (10.77)
Finally, we apply (10.43) to find the relativistic mapping:
P2
u
E/c2 +M +
k
2
u2 = − Ax
E/c2 +m
+ α,
PuPu
E/c2 +M +
k
2
uv = − Ay
E/c2 +m
,
P2
v
E/c2 +M +
k
2
v2 =
Ax
E/c2 +m
+ α,
L = 2L. (10.78)

Chapter 11
Conclusions
In this thesis, we have studied the geometry of a given system’s parameter space from
classical and quantum points of view. We did this with the aid of the quantum met-
ric tensor and the Berry curvature on the quantum side, and with the classical metric
and the Hannay curvature on the classical side. Different formulations of these impor-
tant geometric objects have been presented and have been thoroughly discussed and
exemplified, showing their relevance in the study of quantum and classical systems.
In Chapter 2, we mainly described the basic elements of the parameter space geom-
etry, i.e., the quantum metric tensor and the Berry curvature. Our contribution here
was the proof of the equivalence between the original Hilbert space formulation of the
quantum geometric tensor (2.25) and the path integral approach (2.58). The proof is
fundamental because it puts on a solid ground this approach and opens its confident ap-
plicability to different quantum systems. As a byproduct, we saw that the path integral
approach can be used for a general quantum state, not only for the ground state.
Chapter 3 contains an important addition to the literature of the quantum geometric
tensor. We presented a novel formulation in the phase space that accounts for the Abelian
and non-Abelian quantum geometric tensor. The phase space function Ai(q, p; x) was
shown to possess all the relevant information of the parameter space, and remarkably, it
was seen that to build the quantum metric tensor we only require the Wigner functions.
Chapter 4 is at the heart of this thesis. We introduced for the first time the classical
analog of the quantum metric tensor with the simple, but crucial observation, that the
displacements of the parameters in the phase space coordinates are canonically gener-
ated. In this way, we established a parallelism with the quantum expressions, and thus,
proposed the classical metric. We also contributed with the introduction of the time-
dependent approach to the Hannay curvature and the classical metric, and showed how
both of them emerged from the semiclassical approximation of their quantum counter-
parts. Here, too, we provided the proof of the equivalence of both approaches, which
validates our initial proposal of the classical metric.
In the examples presented in Chapters 5 and 6, we explored a wide range of systems
to get acquainted with the quantum and classical methods and established a comparison
between them. We particularly highlight the quartic anharmonic oscillator and the spin-
167
168 Conclusions
half particle in an external field. In the first case, we proposed an algorithm to adapt
canonical perturbation theory to the computation of the classical metric and found a
good agreement with the quantum metric. As for the second case, the use of Grassmann
variables allowed us to reproduce the quantum results.
The application of the geometric methods to the study of quantum phase transitions
in the Dicke and Lipkin-Meshkov-Glick models was carried out in Chapters 7, 8 and 9.
The analysis is especially relevant, because it shows that the classical description of the
parameter space geometry is powerful and contains all, or almost all the elements as its
quantum counterpart. For the Dicke model, the classical metric and its scalar curvature
were obtained for the first time. In the Lipkin-Meshkov-Glick model, we obtained the
quantum and classical metrics in the thermodynamic limit, and showed that from the
finite-size analysis, it can be inferred that the scalar curvature does not diverge at the
critical point, although it clearly indicates a precursor of the quantum phase transition.
Finally, in Chapter 10, our contribution was the construction of the Higgs mechanism
on the N = 4 super Yang-Mills theory that accounts for the emergence of a hydrogen-like
relativistic theory. We saw that the hidden SO(4) symmetry in this system is restored,
and through the introduction of a relativistic Runge-Lenz vector, we found the quantum
and classical solutions. We also built the infinitesimal canonical transformations gener-
ated by the relativistic Runge-Lenz vector and we were able to reconstruct the spectrum
of the system by means of a natural identification of relativistic and non-relativistic
quantities. As an additional contribution, we employed the Kustaanheimo-Stiefel trans-
formation to relate the spectra of the modified relativistic harmonic oscillator and the
modified relativistic Coulomb problem, thus promoting the duality between these sys-
tems to the relativistic realm.
Many interesting lines for future research are opened with this work. For instance,
the application of the path integral formulation to excited states may have interesting
implications in field theory which are worth studying. Regarding the phase space for-
mulation, a further analysis of the quantum geometric tensor might shed light on the
parameter space of many-body systems and quantum optics models where the Wigner
function formulation is heavily used, and in this sense, it may be useful in the quest
for chaos indicators. As often seen along this work, the classical metric contains all,
or almost all the information of the parameter space of a given system. Therefore, the
classical methods can be used in a variety of many-body models by employing different
tools such as coherent states, Grassmann variables and perturbation theory with the
hope that the analysis of the deviations from the quantum results can help understand
how quantum corrections arise. Naturally, the geometric analysis is not restricted to the
metric tensor only and more elements such as its geodesics, its Riemann tensor, and a
variety of scalars in higher-dimensional parameter spaces can be further examined with
the hope that a connection with the phenomenon of entanglement, which is of great
relevance in quantum information, can be found.
Appendices
169

Appendix A
Classical generators for the quartic
anharmonic oscillator
In this appendix, we give the expression for the generators Gi of the quartic anharmonic
oscillator (see Section 5.2) with Hamiltonian
H =
p2
2m
+
k
2
q2 +
λ
4!
q4. (A.1)
The parameters are x = {xi} = (m, k, λ), i = 1, 2, 3. To order O(λ2), the generator Gi
has the form
Gi(φ0, I; x) = αi0 + αi1λ+ αi2λ
2, (A.2)
where in the case i = 1:
α10 =− I sinφ0 cosφ0
2m
,
α11 =− I2 sin3 φ0 cosφ0(2 cosφ0 − 3)
48
√
m3k3
,
α12 =− I3 cosφ0(647 sinφ0 − 329 sin 3φ0 + 125 sin 5φ0 − 30 sin 7φ0 + 3 sin 9φ0)
27648m2k3
, (A.3)
while for i = 2:
α20 =− I sinφ0 cosφ0
2k
,
α21 =− I2 sin 2φ0(7 cos 2φ0 − cos 4φ0 − 12)
192
√
mk5
,
α22 =− I3 cosφ0(583 sinφ0 − 221 sin 3φ0 + 65 sin 5φ0 − 12 sin 7φ0 + sin 9φ0)
9216mk4
, (A.4)
171
172 Classical generators for the quartic anharmonic oscillator
and for i = 3:
α30 =− I2 cosφ0(9 sinφ0 − sin 3φ0)
96
√
mk3
,
α31 =
I3 cosφ0(551 sinφ0 − 167 sin 3φ0 + 35 sin 5φ0 − 3 sin 7φ0)
13824mk3
,
α32 =
I4 sin 2φ0(4547 cos 2φ0 − 1696 cos 4φ0 + 446 cos 6φ0 − 72 cos 8φ0 + 5 cos 10φ0 − 5480)
884736
√
m3k9
.
(A.5)
Appendix B
Grassmann variables
The Grassmann variables are anticommuting numbers. In physics, they appear in the
path integral formulation of a fermionic field [53]. When quantizing a classical theory,
one promotes the phase space variables (q, p) to operators (q̂, p̂) acting on states |ψ〉 that
belong to a Hilbert space. Additionally, the fundamental Poisson brackets
{qa, qb}P = 0, {pa, pb}P = 0, {qa, pb}P = δab (B.1)
are promoted to commutation relations through the replacement {·, ·}P → 1
i~
[·, ·], which
means that
[q̂a, q̂b] = 0, [p̂a, p̂b] = 0, [q̂a, p̂b] = i~δab . (B.2)
Nevertheless, this process applies only to bosonic degrees of freedom. In the case of
fermions, we must take into account the Pauli exclusion principle, and as a consequence,
we must impose the anticommutation relations
{q̂a, q̂b} = 0, {p̂a, p̂b} = 0, {q̂a, p̂b} = i~δab . (B.3)
The classical version of such a theory requires anticommuting numbers, thus, the product
between two Grassmann numbers θ and θ′ satisfies
θθ′ = −θ′θ, (B.4)
which implies that θ2 = 0. If we have a set of n generators θ = {θ1, ...θn} that satisfy
the anticommutation relations
{θi, θj} = 0, (i, j = 1, ..., n), (B.5)
then the set of linear combinations of θ with c-number coefficients (usual commuting
numbers) is called a Grassmann number, and the algebra generated by θ is called the
Grassmann algebra Λn. Therefore, a function f ∈ Λn has the expansion
f(θ) = f0 +
n
∑
i=1
fiθi +
∑
i<j
fijθiθj + ...+
∑
i<j<k
fijkθiθjθk + ...+ f1···nθ1...θn, (B.6)
173
174 Grassmann variables
where the coefficients fij··· are completely antisymmetric. Of course, a Grassmann num-
ber θi commutes with a c-number.
Two types of linear operators can be introduced to differentiate a function f(θ): the
left derivative
→
∂
∂θi
f(θ) and the right derivative f(θ)
←
∂
∂θi
. Their action on the generators
yields
→
∂
∂θi
θj = δij, θj
←
∂
∂θi
= δij. (B.7)
The derivatives anticommute with θi, thus the Leibniz rule takes the form
→
∂
∂θi
(θjθk) =
→
∂θj
∂θi
θk − θj
→
∂θk
∂θi
= δijθk − δikθj, (B.8)
and analogously for the right derivative. For higher orders, the following relations hold:
→
∂
∂θi
→
∂
∂θj
+
→
∂
∂θj
→
∂
∂θi
= 0,
←
∂
∂θi
←
∂
∂θj
+
←
∂
∂θj
←
∂
∂θi
= 0. (B.9)
If besides the n generators θ of the Grassmann algebra Λn, we add n more generators
denoted as θ∗ and introduce an involution ∗, then the resulting algebra Λ2n has the
properties
{θi, θ∗j} = 0,
(θ∗i )
∗ = θi,
(θiθj)
∗ = θ∗j θ
∗
i ,
(fi1···inθi1 · · · θin)∗ = f ∗i1···inθ
∗
in · · · θ∗i1 , (B.10)
where f ∗i1···in is the complex conjugate of fi1···in .
Appendix C
Bloch coherent states
The Bloch coherent states (or SU(2) coherent states) [139, 140] can be defined through
a displacement operator parameterized by a complex number ζ. In units where ~ = 1,
the displacement operator is
Ω̂(ζ) = eζĴ+−ζ
∗Ĵ− . (C.1)
The form of this operator is reminiscent of the corresponding displacement operator for
the harmonic oscillator coherent states. In the case of angular momentum, the role of
the creation and annihilation operators is played by Ĵ+ and Ĵ−, respectively. Choosing
the parameterization ζ = θ
2
e−iφ, it is easy to see that the operator Ω̂(ζ) takes the form
Ω̂(ζ) = e−iθ(Ĵx sinφ−Ĵy cosφ), (C.2)
which represents a rotation by an angle θ around the axis n = (sinφ,− cosφ, 0).
Now, using the disentangling theorem for angular momentum operators [140], we can
recast Ω̂(ζ) in a more useful form as
Ω̂(ζ) = eζĴ+−ζ
∗Ĵ− = ezĴ+eln(1+|z|
2)Ĵze−z
∗Ĵ− , (C.3)
where, if ζ = θ
2
e−iφ, then z is parameterized as z = e−iφ tan θ
2
. Following the convention of
reference [140], a Bloch coherent state |z〉 is defined by the application of the displacement
operator (C.1) to the state with the lowest angular momentum projection, i.e.,
|z〉 = Ω̂(ζ)|j,−j〉 = ezĴ+eln(1+|z|
2)Ĵze−z
∗Ĵ− |j,−j〉. (C.4)
We can get rid of two operators by noticing that the only contribution in the series
expansion of the rightmost exponential is the identity. Thus, we are left with
|z〉 = ezĴ+eln(1+|z|
2)Ĵz |j,−j〉
= ezĴ+e−j ln(1+|z|
2)|j,−j〉
=
1
(1 + |z|2)j
ezĴ+ |j,−j〉. (C.5)
175
176 Bloch coherent states
Expanding the remaining exponential and using
Ĵk+|j,−j〉 =
√
k!(2j)!
(2j − k)!
|j,−j + k〉, k = 0, 1, ..., 2j , (C.6)
we finally arrive at the expression of the Bloch coherent states in terms of |j,m〉 states:
|z〉 = 1
(1 + |z|2)j
j
∑
m=−j
(
2j
j +m
)1/2
zj+m|j,m〉. (C.7)
With this equation at hand, it is not hard to show that the overlap between two Bloch
coherent states is
〈z|z′〉 = (1 + z∗z′)2j
(1 + |z|2)j(1 + |z′|2)j , (C.8)
which implies that 〈z|z〉 = 1. Furthermore, their completeness relation has the form
2j + 1
4π
∫
dθdφ sin θ |z〉〈z| = 1̂, (C.9)
where the parameterization z = e−iφ tan θ
2
has been used.
One of the most important features of the Bloch coherent states is that the expecta-
tion values of the angular momentum components take on a simple and intuitive form.
Let us first consider the expectation value of Ĵz. Using (C.7), we get
〈z|Ĵz|z〉 =
1
(1 + |z|2)2j
j
∑
m′=−j
(
2j
j +m′
)1/2
z∗j+m
′
j
∑
m=−j
(
2j
j +m
)1/2
zj+m〈j,m′|Ĵz|j,m〉
=
1
(1 + |z|2)2j
j
∑
m=−j
(
2j
j +m
)
m|z|2(j+m)
=
1
(1 + |z|2)2j
2j
∑
k=0
(
2j
k
)
k|z|2k − j
1
(1 + |z|2)2j
2j
∑
k=0
(
2j
k
)
|z|2k
=
|z|2
(1 + |z|2)2j
∂
∂(|z|2)
2j
∑
k=0
(
2j
k
)
|z|2k − j
1
(1 + |z|2)2j
2j
∑
k=0
(
2j
k
)
|z|2k. (C.10)
Recognizing the binomial expansion
2j
∑
k=0
(
2j
k
)
|z|2k = (1 + |z|2)2j, (C.11)
we arrive at
〈z|Ĵz|z〉 =
|z|2
(1 + |z|2)2j (2j)(1 + |z|2)2j−1 − j
= −j 1− |z|2
1 + |z|2 . (C.12)
177
Proceeding in an analogous fashion with Ĵ+ and Ĵ−, we find that
〈z|Ĵ+|z〉 = j
2z∗
1 + |z|2 ,
〈z|Ĵ−|z〉 = j
2z
1 + |z|2 ,
〈z|Ĵx|z〉 = j
2Rez
1 + |z|2 ,
〈z|Ĵy|z〉 = −j 2Imz
1 + |z|2 ,
〈z|Ĵz|z〉 = −j 1− |z|2
1 + |z|2 . (C.13)
which upon using the parameterization z = e−iφ tan θ
2
can be recast as
〈z|Ĵ+|z〉 = jeiφ sin θ,
〈z|Ĵ−|z〉 = je−iφ sin θ,
〈z|Ĵx|z〉 = j sin θ cosφ,
〈z|Ĵy|z〉 = j sin θ sinφ,
〈z|Ĵz|z〉 = −j cos θ. (C.14)
These expressions allow us to define the classical angular momentum variables Jx,Jy,
and Jz as the corresponding expectation values 〈z|Ĵx|z〉, 〈z|Ĵy|z〉, and 〈z|Ĵz|z〉. Thus,
we see that J2
x +J
2
y +J
2
z = j2. The canonical coordinates can be found by requiring that
(Jx, Jy, Jz) satisfy the SU(2) algebra
{Ji, Jj} = iǫijkJk. (C.15)
It turns out that the pair (φ,−j cos θ) = (φ, Jz) is canonical in this sense, therefore,
writing the angular momentum variables as
Jx =
√
j2 − J2
z cosφ, Jy =
√
j2 − J2
z sinφ, (C.16)
we can readily see that the Poisson brackets
{Ji, Jj} =
∂Ji
∂φ
∂Jj
∂Jz
− ∂Ji
∂Jz
∂Jj
∂φ
(C.17)
reproduce the relations (C.15). The canonical pair (φ, Jz) can be thought of as the action-
angle variables of the classical Hamiltonian Hcl that corresponds to Ĥ ∝ Ĵz, which in
terms of the canonical coordinates Q =
√
2(j + Jz) cosφ, P = −
√
2(j + Jz) sinφ has
the form of the harmonic oscillator, i.e., Hcl ∝ 1
2
(P 2 +Q2).

Appendix D
Wolfram Mathematica codes
Here, we show the codes to compute the quantum metric tensor in the anisotropic LMG
model (see Chapter 8) for a given j and a specified mesh in h and γ . The code was first
run using a kernel for each metric component, then the determinant was computed, and
finally, the scalar curvature.
D.1 Quantum metric tensor
In[ ]:= SetDirectoryNotebookDirectory[];
j = 20;
Jp = IdentityMatrix2 j + 1;
Jm = IdentityMatrix2 j + 1;
Jz = IdentityMatrix2 j + 1; DoDoJzj - m + 1, j - mp + 1 = mp KroneckerDelta[m, mp] , mp, j, -j, -1, m, j, -j, -1
DoDoJpj - m + 1, j - mp + 1 =
√j - mp j + mp + 1 KroneckerDelta[m, mp + 1], mp, j, -j, -1, m, j, -j, -1
DoDoJmj - m + 1, j - mp + 1 =
√j + mp j - mp + 1 KroneckerDelta[m, mp - 1], mp, j, -j, -1, m, j, -j, -1
Jx = Simplify[(1/ 2) (Jp + Jm)];
Jy = Simplify[(1/(2 I)) (Jp - Jm)];
H = Simplify-2 h Jz - 1 j (Jx.Jx + γ Jy.Jy);
O1 = -2 Jz;
O2 = -1 j (Jy.Jy);
Quantum metric for the ground state
In[ ]:= G11ground[x_, y_] := Module{Hnum, system, Ener, vec}, Hnum = H /. {h → x, γ → y};
system = TransposeSortByTransposeEigensystem[Hnum], First;
Ener = Part[system, 1];
vec[m_] := Part[system, 2, m];
Sum(vec[1].O1.vec[m])2 (Ener[[m]] - Ener[[1]])2, m, 2, 2 j + 1
G12ground[x_, y_] := Module{Hnum, system, Ener, vec}, Hnum = H /. {h → x, γ → y};
system = TransposeSortByTransposeEigensystem[Hnum], First;
Ener = Part[system, 1];
vec[m_] := Part[system, 2, m];
Sum((vec[1].O1.vec[m]) (vec[m].O2.vec[1]))(Ener[[m]] - Ener[[1]])2, m, 2, 2 j + 1
G22ground[x_, y_] := Module{Hnum, system, Ener, vec}, Hnum = H /. {h → x, γ → y};
system = TransposeSortByTransposeEigensystem[Hnum], First;
Ener = Part[system, 1];
vec[m_] := Part[system, 2, m];
Sum(vec[1].O2.vec[m])2 (Ener[[m]] - Ener[[1]])2, m, 2, 2 j + 1
179
180 Wolfram Mathematica codes
h ∈ [ai,af] with step size Δ1γ ∈ [bi,bf] with step size Δ2
ai = 0.6;
af = 1.2;
bi = -0.6;
bf = -0.4;Δ1 = 0.001;Δ2 = 0.001;
hline = Tablem, m, ai, af, Δ1;γline = Tablem, m, bi, bf, Δ2;
G11 = Array0 &, Lengthhline× Lengthγline, 3;
i = 1;
DoDoPartG11, i = hline[[m]], γline[[n]], G11groundhline[[m]], γline[[n]];
i++, n, 1, Lengthγline, m, 1, Lengthhline // AbsoluteTiming
Export"G11_j20.csv", G11
In[ ]:= ClearJp, Jm, Jx, Jy, Jz, H, O1, O2, hline, γline, G11
D.2 Determinant
SetDirectoryNotebookDirectory[];
datG11 = Import"G11_j20.csv";
datG12 = Import"G12_j20.csv";
datG22 = Import"G22_j20.csv";
Number of points in h
N1 = 601;
Number of points in γ
N2 = 201;
Determinant
G = Array[0 &, {N1 N2, 3}];
DoPartG, i = datG11i, 1, datG11i, 2, datG11i, 3* datG22i, 3 - datG12i, 32, i, 1, N1 N2
Export"G_j20.csv", G
In[ ]:= Clear[datG11, datG12, datG22, G]
Scalar curvature 181
D.3 Scalar curvature
SetDirectory[NotebookDirectory[]];
datg11 = Import["G11_j20.csv"];
datg12 = Import["G12_j20.csv"];
datg22 = Import["G22_j20.csv"];
datg = Import["G_j20.csv"];
h ∈ [ai,af] with step size Δ1γ ∈ [bi,bf] with step size Δ2
ai = 0.6;
af = 1.2;
bi = -0.6;
bf = -0.4;Δ1 = 0.001;Δ2 = 0.001;
hline = Table[m, {m, ai, af, Δ1}];γline = Table[m, {m, bi, bf, Δ2}];
Converts the metric and determinant from 3-column format to mesh format
In[ ]:= g11 = Array[0 &, {Length[hline], Length[γline]}];
g12 = Array[0 &, {Length[hline], Length[γline]}];
g22 = Array[0 &, {Length[hline], Length[γline]}];
g = Array[0 &, {Length[hline], Length[γline]}];
Ain = Array[0 &, {Length[hline], Length[γline]}];
Bin = Array[0 &, {Length[hline], Length[γline]}];
R = Array[0 &, {Length[hline], Length[γline]}];
In[ ]:= k = 1;
Do[Do[
g11[[i, j]] = datg11[[k, 3]];
g12[[i, j]] = datg12[[k, 3]];
g22[[i, j]] = datg22[[k, 3]];
g[[i, j]] = datg[[k, 3]];
k++, {j, 1, Length[γline]}], {i, 1, Length[hline]}]
Clear[k]
182 Wolfram Mathematica codes
Computation of the scalar curvature
In[ ]:= Der10G11 = NDSolve`FiniteDifferenceDerivative[{1, 0}, {hline, γline}, g11];
Der01G11 = NDSolve`FiniteDifferenceDerivative[{0, 1}, {hline, γline}, g11];
Der10G12 = NDSolve`FiniteDifferenceDerivative[{1, 0}, {hline, γline}, g12];
Der10G22 = NDSolve`FiniteDifferenceDerivative[{1, 0}, {hline, γline}, g22];
In[ ]:= DoDoAin[[i, j]] = 1 √g[[i, j]] g12[[i, j]] g11[[i, j]] Der01G11[[i, j]]- Der10G22[[i, j]];
Bin[[i, j]] = 1 √g[[i, j]] 2 Der10G12[[i, j]]- Der01G11[[i, j]]-g12[[i, j]] g11[[i, j]] Der10G11[[i, j]], {j, 1, Length[γline]},
{i, 1, Length[hline]}
In[ ]:= A = NDSolve`FiniteDifferenceDerivative[{1, 0}, {hline, γline}, Ain];
B = NDSolve`FiniteDifferenceDerivative[{0, 1}, {hline, γline}, Bin];
In[ ]:= DoDoR[[i, j]] = 1 √g[[i, j]] (A[[i, j]]+ B[[i, j]]), {j, 1, Length[γline]}, {i, 1, Length[hline]}
In[ ]:= Clear[g11, g12, g22, g, Ain, Bin, Der10G11, Der01G11, Der10G12, Der10G22, A, B]
Converts the scalar curvature from mesh format to 3-column format
In[ ]:= datR = Array[0 &, {Length[hline]× Length[γline], 3}];
In[ ]:= k = 1;
Do[Do[
Part[datR, k] = {hline[[i]], γline[[j]], R[[i, j]]}; k++, {j, 1, Length[γline]}], {i, 1, Length[hline]}]
Clear[k]
In[ ]:= Clear[R]
Export["R_j20.csv", datR]
In[ ]:= Clear[datg11, datg12, datg22, datg, datR]
Appendix E
Quantum geometric tensor for Bloch
coherent states in the modified LMG
model
In this section, we compute the quantum geometric tensor for the modified LMG model
(see Chapter 9) using Bloch coherent states. These states are parameterized by the
complex number z = e−iφ tan θ
2
and are given by
|z〉 = ezĴ+ |j,−j〉
(1 + |z|2)j =
j
∑
m=−j
c(j)m |j,m〉, (E.1)
where
c(j)m :=
(
2j
j +m
)1/2
sinj+m
θ
2
cosj−m
θ
2
e−i(j+m)φ. (E.2)
The angles on the Bloch sphere are functions of the parameters, i.e., θ = θ(x) and
φ = φ(x), with x = {xi} = (Ωx, ξy), i = 1, 2. The coherent state |z〉 that corresponds to
the highest energy state has the coordinates:
• (θ4, φ4) =
(
arccos
(
− 1
2ξy
)
, arccos
(
Ωx√
4ξ2y−1
))
for ξy >
√
Ω2
x+1
2
.
• (θ1, φ1) =
(
arccos
(
− 1√
1+Ω2
x
)
, 0
)
for ξy ≤
√
Ω2
x+1
2
.
The quantum geometric tensor for the coherent state |z〉 is given by
G
(z)
ij = 〈∂iz|∂jz〉 − 〈∂iz|z〉〈z|∂jz〉 =
j
∑
m=−j
∂ic
∗(j)
m ∂jc
(j)
m −
j
∑
m=−j
∂ic
∗(j)
m c(j)m
j
∑
n=−j
c∗(j)n ∂jc
(j)
n .
(E.3)
183
184 Quantum geometric tensor for Bloch coherent states in the modified LMG model
With this at hand, we find the metric tensor, its determinant, and the Berry curvature
for both phases.
-For ξy >
√
Ω2
x+1
2
(broken phase):
F
(z)
12 = −j
2
1
ξ2y
√
4ξ2y − Ω2
x − 1
,
g
(z)
11 =
j
2
4ξ2y − 1
4ξ2y(4ξ
2
y − Ω2
x − 1)
,
g
(z)
12 = −j
2
Ωx
ξy(4ξ2y − Ω2
x − 1)
,
g
(z)
22 =
j
2
Ω2
x + 1
ξ2y(4ξ
2
y − Ω2
x − 1)
,
g(z) =
j2
16ξ4y(4ξ
2
y − Ω2
x − 1)
. (E.4)
-For ξy <
√
Ω2
x+1
2
(symmetric phase):
F
(z)
12 = 0,
g
(z)
11 =
j
2
1
(Ω2
x + 1)2
,
g
(z)
12 = 0,
g
(z)
22 = 0,
g(z) = 0. (E.5)
Since for the symmetric phase, ξy <
√
Ω2
x+1
2
, the determinant of the metric is zero,
the scalar curvature is not defined there. On the other hand, in the broken phase,
ξy >
√
Ω2
x+1
2
, it is possible to compute it using (1.20), which yields
R = 4/j. (E.6)
In Figure E.1, we plot the resulting quantum metric tensors and their scalar curvature.
In [5], the authors found the scalar curvature for the Bloch coherent states taking (θ, φ)
as parameters, obtaining the same result (E.6). Due to the invariant nature of R under
coordinate transformations, we conclude that the metric (E.4) is actually the metric of
a sphere (see Chapter 1) in more complicated coordinates.
185
0 1 2 3 4 5 6
Ωx0
2000
4000
6000
8000
g11
0 1 2 3 4 5 6
Ωx0.1
1
10
100
1000
104
g11
H-P Numeric Coherent
H-P
Numeric
Coherent
1 2 3 4 5 6
Ωx
-30
-25
-20
-15
-10
-5
0
g12
(a) (b)
H-P
Numeric
Coherent
0 1 2 3 4 5 6
Ωx0
10
20
30
40
50
60
g22
H-P
Numeric
Coherent
1 2 3 4 5 6
Ωx
-4
-3
-2
-1
0
R
(c) (d)
Figure E.1: Comparison of the quantum metric tensor and the scalar curvature obtained with the
truncated Holstein-Primakoff approximation (solid black), coherent states (dot-dashed orange), and
exact diagonalization (dashed blue). We fixed j = 96 and ξy = 2.3.

Appendix F
Precision considerations in the
modified LMG model
In this section, our aim is to give some details regarding precision requirements for the
computation of the quantum metric tensor in the modified LMG model (see Chapter
9). We only explore the energy levels, which is sufficient to provide insight into the
phenomenon. To that end, consider the following Hamiltonian
Ĥ∗ = ΩxĴx +
ξy
j
Ĵ2
y . (F.1)
This Hamiltonian has an exact degeneracy when Ωx = 0, in contrast to the LMG Hamil-
tonian (9.1). In Figure F.1, we show the energy of the four upper eigenstates for both
Hamiltonians. We notice that the energy levels begin getting together near the critical
Ωx where the quantum phase transition takes place in the thermodynamic limit.
To further characterize the level crossing for both Hamiltonians, we take the logarithm
of the difference in energies between the maximum state (n = 2j+1) and the state closest
to the maximum (n = 2j). We show in Figure F.2 the plots for j = 32 and ξy = 2.3.
Some crucial features can be observed from the plots. As Figure F.2 (a) and (c) clearly
show, machine precision of 15 decimal digits is not enough to have a good resolution of
the energy difference; as a consequence, the computation of the quantum metric tensor
is affected, since the denominator of equation (2.64) blows up. Now, if we raise the
machine precision to a digit accuracy of 601, then Figure F.2 (b) and (d) are obtained.
We thus see that the energy difference of the two upper levels of the LMG Hamiltonian
(9.1) is now resolved, as opposed to the Hamiltonian Ĥ∗, which continues unresolved
due to its exact degeneracy. This brief analysis shows the importance of incorporating
digit accuracy in the numerical computations, otherwise, the results would show a high
degree of noise and conclusions could not be extracted with certainty. Naturally, as j
increases, the precision requirement and the computing time is higher. This restricts the
possibility to study higher values of j.
1The Wolfram Mathematica 12.1 software was used to undertake this task.
187
188 Precision considerations in the modified LMG model
-5 5
Ωx
2
4
6
8
〈H LMG〉/j
2j+1 2j 2j-1 2j-2
-5 5
Ωx
2
4
6
8
〈H *〉/j
2j+1 2j 2j-1 2j-2
(a) (b)
Figure F.1: Expectation value of ĤLMG taken in the upper four eigenstates (a), and of Ĥ∗ (b). We
fixed j = 32 and ξy = 2.3. The levels begin getting together near Ωxc = 4.490.
189
-5 5
Ωx
-15
-10
-5
log10(ΔELMG/j)
-5 5
Ωx
-35
-30
-25
-20
-15
-10
-5
log10(ΔELMG/j)
(a) (b)
-5 5
Ωx
-15
-10
-5
log10(ΔE*/j)
-5 5
Ωx
-50
-40
-30
-20
-10
log10(ΔE*/j)
(c) (d)
Figure F.2: Energy difference between the states n = 2j+1 and n = 2j (blue) and between the states
n = 2j− 1 and n = 2j− 2 (black) for the Hamiltonian ĤLMG ((a) and (b)) and for the Hamiltonian Ĥ∗
((c) and (d)). We fixed j = 32 and ξy = 2.3. The Hamiltonian Ĥ∗ presents a level crossing at Ωx = 0,
in contrast to ĤLMG, which has an avoided crossing there. The plots in the left column were computed
with machine precision, whereas those in the right have a precision of 60 decimal digits.

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