UNIVERSIDAD NACIONAL AUTÓNOMA DE MÉXICO POSGRADO EN CIENCIAS FÍSICAS INSTITUTO DE CIENCIAS NUCLEARES FÍSICA DE ALTAS ENERGÍAS, FÍSICA NUCLEAR, GRAVITACIÓN Y FÍSICA MATEMÁTICA MÉTODOS GEOMÉTRICOS Y ALGEBRAICOS EN INFORMACIÓN CUÁNTICA. TESIS QUE PARA OPTAR POR EL GRADO DE: DOCTOR EN CIENCIAS: FÍSICA PRESENTA: JOAN AITOR AUSTRICH OLIVARES TUTOR PRINCIPAL DR. JOSÉ DAVID VERGARA OLIVER. MIEMBROS DEL COMITÉ TUTOR DR. JOSÉ ANTONIO RAFAEL GARCÍA ZENTENO DR. MARIANO CHERNICOFF MINSBERG. CIUDAD DE MÉXICO, MÉXICO, JULIO DE 2025 1 UNAM – Dirección General de Bibliotecas Tesis Digitales Restricciones de uso DERECHOS RESERVADOS © PROHIBIDA SU REPRODUCCIÓN TOTAL O PARCIAL Todo el material contenido en esta tesis esta protegido por la Ley Federal del Derecho de Autor (LFDA) de los Estados Unidos Mexicanos (México). El uso de imágenes, fragmentos de videos, y demás material que sea objeto de protección de los derechos de autor, será exclusivamente para fines educativos e informativos y deberá citar la fuente donde la obtuvo mencionando el autor o autores. Cualquier uso distinto como el lucro, reproducción, edición o modificación, será perseguido y sancionado por el respectivo titular de los Derechos de Autor. PROTESTA UNIVERSITARIA DE INTEGRIDAD Y HONESTIDAD ACADÉMICA Y PROFESIONAL (Graduación con trabajo escrito) De conformidad con lo dispuesto en los artículos 87, fracción V, del Estatuto General, 68, primer párrafo, del Reglamento General de Estudios Universitarios y 26, fracción I, y 35 del Reglamento General de Exámenes, me comprometo en todo tiempo a honrar a la Institución y a cumplir con los principios establecidos en el Código de Ética de la Universidad Nacional Autónoma de México, especialmente con los de integridad y honestidad académica. De acuerdo con lo anterior, manifiesto que el trabajo escrito titulado: que presenté para obtener el grado de es original, de mi autoría y lo realicé con el rigor metodológico exigido por mi programa de posgrado, citando las fuentes de ideas, textos, imágenes, gráficos u otro tipo de obras empleadas para su desarrollo. En consecuencia, acepto que la falta de cumplimiento de las disposiciones reglamentarias y normativas de la Universidad, en particular las ya referidas en el Código de Ética, llevará a la nulidad de los actos de carácter académico administrativo del proceso de graduación. Atentamente (Nombre, firma y Número de cuenta de la persona alumna) Métodos Geométricos y Algebraicos en Información Cuántica. Joan Aitor Austrich Olivares 307665445 ----Doctorado---- 2 COORDINACIÓN GENERAL DE ESTUDIOS DE POSGRADO CARTA AVAL PARA DAR INICIO A LOS TRÁMITES DE GRADUACIÓN Universidad Nacional Autónoma de México Secretaría General Coordinación General de Estudios de Posgrado Dr. Alberto Güijosa Hidalgo Programa de Posgrado en Ciencias Físicas Presente Quien suscribe, , tutor(a) principal de , con número de cuenta , integrante del alumnado de de ese programa, manifiesto bajo protesta de decir verdad que conozco el trabajo escrito de graduación elaborado por dicha persona, cuyo título es: , así como el reporte que contiene el resultado emitido por la herramienta tecnológica de identificación de coincidencias y similitudes con la que se analizó ese trabajo, para la prevención de faltas de integridad académica. De esta manera, con fundamento en lo previsto por los artículos 96, fracción III del Estatuto General de la UNAM; 21, primero y segundo párrafos, 32, 33 y 34 del Reglamento General de Exámenes y; 22, 49, primer párrafo y 52, fracción II del Reglamento General de Estudios de Posgrado, AVALO que el trabajo de graduación presentado se envíe al jurado para su revisión y emisión de votos, por considerar que cumple con las exigencias de rigurosidad académica previstas en la legislación universitaria. Protesto lo necesario, Ciudad Universitaria, Cd. Mx., a de de 202 Tutor(a) principal Dr. José David Vergara Oliver Joan Aitor Austrich Olivares 307665445 Doctorado en Ciencias (Física) MÉTODOS GEOMÉTRICOS Y ALGEBRAICOS EN INFORMACIÓN CUÁNTICA Dr. José David Vergara Oliver 28 abril 5 3 Resumen de la Tesis Doctoral. Ya no es novedad que la geometŕıa del espacio de parámetros y los fenómenos cuánticos tienen una relación muy especial. Esto se ejemplifica desde los primeros indicios de fases geométricas en luz polarizada y la formulación de Berry, donde las propiedades globales de un sistema quedan determinadas por la estructura del espacio de parámetros. Esta relación ha permitido caracterizar las transiciones de fase cuánticas y determinar invariantes topológicos en varios sistemas de materia condensada. En la actualidad, se están explorando nuevas respuestas no lineales en materia condensada que dependen solamente de las propiedades geométricas del espacio de parámetros. Más aún, el estudio de la geometŕıa subyacente del sistema ha permitido el avance en el estudio de la teoŕıa de información cuántica. La extensión de estas ideas a teoŕıa cuántica de campos y sistemas relativistas con esṕın ha permitido caracterizar sistemas como los semimetales de Weyl, buscar transiciones de fase en cromodinámica cuántica (QCD) e, incluso, en teoŕıa de cuerdas para intentar resolver el misterioso “Swampland”, donde teoŕıas con parámetros muy cercanos pueden presentar comportamientos muy diferentes. En este trabajo nos enfocamos en el estudio del Tensor Geométrico Cuántico (QGT) en sistemas curvos, sistemas con esṕın y en teoŕıas cuánticas de campos de dimensión 0+0. Esta tesis doctoral está organizada en cuatro caṕıtulos principales: • En el primero, introducimos el QGT y se explica cómo la parte real, que es el tensor métrico cuántico, cuantifica la distancia infinitesimal entre los estados en el espacio de parámetros, mientras que su parte imaginaria, que es proporcional a la curvatura de Berry, codifica la interferencia de fase en sistemas cuánticos y nos da el número de Chern, que es el causante de la cuantización de la conductividad eléctrica en sistemas que presentan el efecto Hall. • En el segundo caṕıtulo, extendemos el QGT a espacios curvos, donde la métrica del espacio de con- figuración depende expĺıcitamente de los parámetros del sistema. Esta extensión está completamente determinada por las propiedades geométricas del espacio, incorporando el vector de deformación, σρ, que nos permite redefinir la conexión de Berry para preservar la invariancia de norma del QGT exten- dido. Además, presentamos varios ejemplos en donde se evidencian las correcciones geométricas puras por la curvatura del espacio. • En el tercer caṕıtulo trabajamos con sistemas de esṕın- 12 , tanto en la representación de Dirac como en la de Weyl. Definimos un producto interno adecuado para calcular el QGT para un fermión libre y un modelo sencillo de semimetal de Weyl. En este último ejemplo mostramos que la métrica cuántica describe efectivamente un espacio análogo a la esfera de Bloch, mientras que la curvatura de Berry coincide con los resultados conocidos. • El cuarto caṕıtulo está enfocado en la extensión del QGT a teoŕıas cuánticas de campos a través de la integral de trayectoria, pero nos enfocamos en trabajar en el sistema más sencillo de todos: un campo en dimensión 0+0, es decir, el campo representa una variable aleatoria. Este caso se puede resolver anaĺıticamente, presentando ciertas caracteŕısticas similares a los casos de dimensión mayor, pero sobre todo, se confirma que la superposición de vaćıos codifica información geométrica incluso en el caso sin dimensiones espaciales ni temporales. En conclusión, este trabajo busca unificar un enfoque geométrico en mecánica cuántica al extender el QGT a escenarios curvos y relativistas, aśı como mostrar su relevancia en teoŕıas cuánticas de campos. Las aplicaciones que presentamos en cada caṕıtulo establecen una base para futuras investigaciones en mecánica cuántica, materia condensada, teoŕıa cuántica de campos y teoŕıa de la información. 4 Contents Preface. 7 1 Introduction. 9 1.1 Quantum Geometric Tensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.1.1 Berry connection and curvature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.1.2 Quantum Metric Tensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2 Quantum Geometric Tensor: Parameter-dependent Spacetime metric (“Curved space- time”). 15 2.1 Quantum Metric Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1.1 QMT: Quantum Fidelity and Quantum Susceptibility. . . . . . . . . . . . . . . . . . . 17 2.2 Berry Curvature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3 Quantum Geometric Tensor Curved Background. . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4 Examples of QGT in curved background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.4.1 One-dimensional Curved Anharmonic Oscillator. . . . . . . . . . . . . . . . . . . . . . 22 2.4.2 Oscillator with a Morse type potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4.3 Generalized Harmonic Oscillator in curved space. . . . . . . . . . . . . . . . . . . . . . 25 2.4.4 Coupled Anharmonic Oscillator in curved space. . . . . . . . . . . . . . . . . . . . . . 28 3 QGT for spin- 12 particles. 33 3.1 QGT for Dirac free particle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.2 QGT Dirac representation (λ = {m}). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3 QGT Weyl representation (λ = {m}). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.4 Fidelity approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.4.1 Dirac Representation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.4.2 Weyl Representation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.5 QGT for spin- 12 particles with λ⃗ = (m, p3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.6 QGT for a Weyl Semimetal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4 Quantum Geometric Tensor in Quantum Field Theories. 51 4.1 QGT in d+ 1 dimensional QFT’s: general theory. . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.2 QGT in 0 + 0 Dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.3 Quantum Geometric Tensor for the free theory with a source. . . . . . . . . . . . . . . . . . . 56 4.3.1 QGT in 0 + 0 dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.3.2 QGT in 1 + 0 (Quantum Mechanics). . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.3.3 QGT in 1 + 1 dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.3.4 Generalization of the QGT components for any spatial and time dimensions. . . . . . 63 4.3.5 Determinant of the QGT for a scalar free theory with a source. . . . . . . . . . . . . . 67 4.3.6 Ricci scalar of the parameter manifold. . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.4 QGT of two coupled fields in a 0 + 0 QFT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.4.1 Case α > 0: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.4.2 Case for α < 0: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5 5 Conclusions 77 A Complex and Holomorphic Gauge Symmetries. 81 A.1 Complex Harmonic Oscillator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 A.2 Calculation of the Dirac brackets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 References. 85 6 Preface. Over the past forty years, the relation between geometry and quantum mechanics has been the subject of increasing attention, due to the astonishing insights it has produced across various areas of physics. This relationship goes back to 1980, with the seminal work of Provost and Vallée [1], who introduced the definition of the quantum geometric tensor (QGT). This tensor was derived from the idea of finding a notion of distance for the quantum states using the parameters of the system. This complex tensor contains the information given from the quantum metric (real part) of the Hilbert space and the Berry curvature (imaginary part)1. One of their main results was the description of the local quantum geometry of the parameters of the system in terms of the Fubini-Study metric. In 1984, M. V. Berry showed [2] that a quantum system undergoing a cyclic adiabatic evolution of its external parameters acquires an additional phase from a purely geometric origin. This phase is now known as the Berry phase, which can be understood as a holonomy associated with a connection on the parameter space. Its discovery led to an explosion of interest in the geometric structure of quantum mechanics, partic- ularly in condensed matter systems where topological invariants such as Chern numbers became crucial to characterizing physical phenomena. Berry’s work also brought renewed attention to earlier ideas of geometric phase, such as those introduced by Pancharatnam in 1956 [3]. In the following decades, the QGT was consolidated as a powerful tool to characterize both local properties (such as distinguishability and distances between the quantum states) and global features (such as topological phases) of quantum systems. A significant step forward came with the perturbative approach developed by Zanardi et al. [4], which revealed, in a heuristical manner, that the QGT exhibits divergences near quantum critical points where the quantum phase transitions occur, thus capturing the sensitivity of the ground state to infinitesimal changes in the parameters of the quantum system. The imaginary part of the QGT, which is proportional to the Berry curvature, has become a central part in the description of topological phases of quantum matter. These include topological insulators, Weyl semimetals, and quantum systems exhibiting the quantum Hall effect. In such systems, the QGT allows one to compute topological invariants and provides insight into the global structure of the parameter manifold [5]. More recently, it has been shown that the real part, i.e., the quantum metric, plays a key role in the emergence of nonlinear effects in the response theory of quantum systems [6]. In the context of quantum information theory, the quantum metric is the same as the quantum suscep- tibility; thus, it is a measure of how distinguishable two quantum states are in the parameter space [7, 8]. This has allowed for extending the application of the QGT to the detection of structure changes in quantum systems, and even in the exploration of quantum resources for emergent technologies. There has been growing interest in extending the QGT formalism beyond standard quantum mechanics into the realm of quantum field theories (QFTs), including quantum chromodynamics (QCD) [9], string theory [10, 11], as well as La- grangian and classical formulations [12–14]. Other efforts focus on defining QGTs in curved backgrounds and geometrically deformed systems [15]. These generalizations pose technical challenges, such as the ab- sence of the Berry connection in the path integral formalism, but they also offer novel geometric insights for understanding the underlying structure of the theories in question. The present work is structured into four main chapters: Chapter 1 provides the motivation for studying the Quantum Geometric Tensor (QGT), along with a detailed introduction to the fundamental concepts involved. In particular, we present the standard formalism where both the quantum metric tensor and the Berry curvature naturally arise as the real and imaginary 1This phase was discovered years later, so in the original paper the authors only talk about it as the imaginary part of the QGT. 7 parts of the QGT, respectively. Chapter 2 follows the approach introduced in [15], one of the central results of this thesis. There, the QGT is extended to quantum systems defined on curved backgrounds or geometrically deformed configuration spaces, where the metric explicitly depends on external parameters. We present several illustrative examples in one dimension, including: a curved anharmonic oscillator, an oscillator with a Morse-type potential, and a generalized harmonic oscillator in curved space. The first two examples do not exhibit Berry curvature, whereas the third one does. In certain limits, both the quantum metric and Berry curvature reduce to their flat-space counterparts. Additionally, we include a two-dimensional example—a coupled anharmonic oscillator in curved space—which will serve as a stepping stone for the extension of the QGT to quantum field theory in Chapter 4. Chapter 3 explores an extension of the QGT to relativistic spin- 12 systems. We construct a suitable inner product using the continuity equation for the probability current. First, we consider a minimal model with the mass m as the sole parameter, analyzing both Dirac and Weyl representations for particles and antiparticles. We show that both representations yield consistent results. We then extend the parameter space to include both mass m and momentum p3. Additionally, we apply the fidelity approach to these systems and compare it with the QGT formalism, identifying their domains of validity. The chapter concludes with an application to a simple Weyl semimetal model, where the parameters are the components of the crystal momentum k⃗. We show that the quantum metric reveals a structure reminiscent of the Bloch sphere, offering a novel way to characterize quantum states. The Berry curvature, in turn, matches known results from the literature. Chapter 4 presents an extension of the QGT to 0 + 0-dimensional quantum field theories, which may be interpreted as theories of random variables or probability distributions over a single variable. These models, despite their simplicity, are exactly solvable (at least in the examples considered), and can provide useful insights into the structure and challenges of higher-dimensional theories. Finally, in the Appendix, we include a discussion on holomorphic gauge symmetries in complexified configuration spaces. This section complements Chapter 2, as it details how the analytical solutions and eigenfunctions of the quantum systems were obtained, since solving the Schrödinger equation for nontrivial cases can be very difficult or only possible via a perturbative approach. With this structure, we aim to provide a unified perspective on quantum geometry from various physical approaches. Acknowledgements Finally I want to thank SECIHTI for the scholarship granted CVU: 863133 and DGAPA-PAPIIT project IN114225-Geometŕıa de la información cuántica. Joan Aitor Austrich Olivares. April, 2025. 8 Chapter 1 Introduction. One of the main goals in science, especially in physics, is to distinguish between different objects, systems, or theories. A common approach is to vary their parameters and compare how they respond. A remark- able framework for analyzing such behavior emerged from the fusion of differential geometry and quantum mechanics in the seminal paper by Provost and Vallée [1]. In it, they introduced the Quantum Geomet- ric Tensor (QGT), which captures how a quantum state changes in response to infinitesimal variations in external parameters. Before their work, several investigations had already suggested a deep connection between geometry and physical properties, particularly through the appearance of geometric phases: phases acquired by a system due to its geometric or topological evolution, rather than dynamical changes. In 1956, Pancharatnam [3] observed that the interference of polarized light beams could exhibit a phase shift related to the solid angle subtended on the Poincaré sphere. This led to the concept of the Pancharatnam phase, described as the retardation required for two polarization states to interfere maximally. Much later, in 1984, M. V. Berry independently discovered that a quantum system undergoing adiabatic evolution around a closed loop in parameter space acquires an unexpected phase [2]. This phase, now called the Berry phase, was later recognized as the quantum counterpart of Pancharatnam’s geometric phase. To understand Berry’s contribution, we must first introduce the notion of a slowly varying parameter, formalized by the Adiabatic Theorem. This theorem states that if a system is initially in an eigenstate of a Hamiltonian H(λ⃗(t)), and if the parameters λ⃗(t) change sufficiently slowly (i.e., adiabatically), then the system will remain in the instantaneous eigenstate of the Hamiltonian throughout the evolution. Berry showed that under such conditions, the quantum state acquires not only a dynamical phase e−iEt/ℏ, but also a geometric, path-dependent phase eiγ(t), which depends solely on the trajectory in parameter space [16]. The idea of slowly varying a parameter naturally leads us to the concept of phase transitions, which are characterized by abrupt changes in the physical properties of a system when a parameter is tuned between two very close values. There exists a critical point of the varying parameter where such a phase transition occurs. Broadly, phase transitions can be classified into two types: quantum and classical. We define quantum phase transitions (QPTs) as points of non-analyticity in the ground-state energy density at zero temperature (T = 0). These transitions are driven by non-thermal parameters such as pressure or external magnetic fields. The singularities can arise either due to level crossings in the ground- state spectrum or as a result of taking the thermodynamic limit. In contrast, classical phase transitions occur at finite temperatures (T ̸= 0) and are typically accompanied by a divergence in both correlation length and correlation time. This means that the order parameter fluctuates over increasingly large distances, but with increasingly slower dynamics. At the transition point, the characteristic frequency of these fluctuations, denoted ω∗, tends to zero. A quantum system behaves classically when the thermal energy dominates over quantum fluctuations, i.e., when ℏω∗ ≪ kBTc near the critical point, ensuring that all critical fluctuations behave classically. Another way to characterize behavior around a quantum phase transition is to ask how “close” two quantum states are. This question is addressed by the concept of quantum fidelity, which quantifies how much a quantum state changes under an infinitesimal variation of the system’s parameters. To leading order, this quantity is related to the Fubini–Study metric, thus providing a notion of quantum distance. 9 Throughout this discussion, we have implicitly assumed the adiabaticity of the system. That is, the parameters evolve so slowly that the system remains in its instantaneous ground state during the evolution. With this in mind, quantum phase transitions can further be classified based on the continuity of the order parameter at the critical point: • First-order QPT: The order parameter exhibits a discontinuous change. This corresponds to a finite discontinuity in the first derivative of the ground-state energy density with respect to the tuning parameter. • Second-order QPT: Also known as a continuous quantum phase transition, where the order parameter smoothly vanishes at the critical point. These are characterized by continuity in the first derivatives of the energy density but discontinuity in the second derivatives. In this work, we are particularly interested in second-order quantum phase transitions, as they may be predicted by the singular behavior of the determinant of the quantum geometric tensor (QGT). Before continuing with the QGT, let us mention some other physical contexts in which it plays an im- portant role. One notable example is the Aharonov–Bohm effect, where a nonzero vector potential (despite a vanishing magnetic field) creates a phase shift, illustrating the link between holonomy and quantum me- chanics [17]. In recent years, the Berry curvature has become a key tool in condensed matter physics [6]1 to characterize topological insulators, Weyl semimetals, and other quantum systems. Increasing attention has also been given to the real part of the quantum geometric tensor—the quantum metric—which captures the geometry of the parameter manifold, in contrast to the topological nature of the Berry curvature. It has been shown that the motion of electrons can be described in terms of the geometry of the Hilbert space of their wavefunctions. Furthermore, the dipoles of the Berry curvature and the quantum metric contribute to nonlinear transport phenomena. For instance, in the presence of an AC current, the system may exhibit a Hall response or a longitudinal response at doubled frequencies. Depending on the broken symmetries, the frequency of the Hall voltage is proportional either to the Berry curvature dipole (in P- broken but time-reversal T -symmetric systems) or, as shown in antiferromagnets, to the quantum metric dipole (in systems with P and T preserved, but PT broken). In another condensed matter system, the kagome magnet, quantum geometry plays a significant role in the magneto-nonlinear Hall effect and nonlinear longitudinal transport. Thus, such nonlinear features can serve as new probes to detect the spectral, symmetrical, and topological properties of quantum matter [6]. In contrast to quantum phase transitions, which occur at zero temperature, it is believed that the quantum metric controls the transition temperatures of flat-band superconductors. Moreover, the discovery of magic- angle twisted bilayer graphene has enabled the study of geometric contributions to the superfluid weight and transition temperature. For Chern insulators, the stability across the Brillouin zone depends on the homogeneity of both the Berry curvature and the quantum metric. Let us now shift to a seemingly unrelated question: how to measure the “similarity” between two theories or objects. One approach is to consider a differential manifold M and use the notion of a divergence between two points p and q on M [18]. A function is called a divergence if it satisfies: 1. D[p, q] ≥ 0. 2. D[p, q] = 0 if and only if p = q. 3. If p and q are sufficiently close, the Taylor expansion of the divergence in terms of the coordinates xp and xq = xp + dx is given by D[p, q] = 1 2 gij(xp)dx idxj + O(|dx|3), (1.1) where gij are the entries of a positive-definite matrix depending on xp. 1More references therein, for the systems mentioned here. 10 This means that the divergence D endows the differential manifold M with a Riemannian structure.2 Moreover, the units of divergence are the square of a distance. It is important to note that a divergence is not a true distance, since it is not symmetric with respect to p and q: D[p, q] ̸= D[q, p], (1.2) and it does not satisfy the triangle inequality. Still, it represents a measure of dissimilarity between points on M. If we take the negative of the entropy S(p), which is a convex function, and consider normalized probability distributions p(x) and q(x), we obtain the Kullback–Leibler divergence: DKL[p(x), q(x)] = ∫ p(x) ln p(x) q(x) dx, (1.3) where x is a random variable. The KL-divergence, also known as relative entropy, gives a notion of proximity between probability distributions. It quantifies the number of measurements required to distin- guish them and the amount of information lost by renormalization group flows starting from the ultraviolet regime—serving as a measure of “fine-tuning” [10]. To connect the notion of divergence with the quantum geometric tensor, consider a family of probability distributions p(x, λ) defined on a continuous random variable x ∈ R n with parameters λ ∈ M, the differential parameter manifold. Each point λ corresponds to a unique distribution. We can interpret the KL-divergence operationally as how likely samples drawn from p(x, λ) appear to come from p(x, λ′), where p(x, λ) is the true distribution and p(x, λ′) is the probe. For further discussion, see [10, 11,18]. Expanding the KL-divergence to second order in δλ, we obtain: DKL[λ, λ+ δλ] = 1 2 ∫ dµ(x) p(x, λ) ∂ ln p(x, λ) ∂λi ∂ ln p(x, λ) ∂λj δλiδλj , (1.4) where gij(λ) = ∫ dµ(x) p(x, λ) ∂ ln p(x, λ) ∂λi ∂ ln p(x, λ) ∂λj (1.5) is the Fisher metric or information metric. This is the unique (up to a constant) Riemannian metric allowed on a statistical manifold, assigning a statistical distance between nearby probability distributions. It’s important to remark that this metric is independent of the degrees of freedom chosen to express the probability distributions, that is, we can realize any coordinate transformation that does not depend on the parameters λ. This is true, since if the transformation depends on the parameters λ the information metric may change abruptly because it will be characterized a completely different family of probabability distrubtions than the original [10]. We can extend this notion of proximity between probability distributions to the quantum field theory (QFT), which has been already put in practice in String Theory, more precisely, in the “String Landscape” [11]. This extension gives rise to the quantum metric which is singular in critical quantum points, which qualitatively, make different scale-invariant predictions when they are compared with other theories within the same family. That is, it indicates the inflection point where the behaviour of the theories are completely different even when the parameters are very close. Thus giving rise to a quantum phase transition, which has been discussed above. In the following section, we will construct the quantum geometric tensor, the object that contains both structures considered in this introduction, the quantum metric and the Berry curvature. 1.1 Quantum Geometric Tensor. The quantum geometric tensor (QGT) is a complex tensor that encodes both geometric and topological properties of the parameter space of a quantum system. This tensor can be defined rigorously using the differential geometry in quantum mechanics. Thus, it provides a geometric framework to study quantum states that explicitly depend on the external parameters of the system. 2Recall that a manifold M is a Riemannian manifold if there exists a positive-definite matrix g(x) such that local distances between nearby points are defined through it. 11 1.1.1 Berry connection and curvature. Let’s consider a quantum system whose dynamics depends on the external parameters λ = (λ1, λ2, ..., λn) ∈ M, where M is an n-dimensional parameter manifold. At each point λ ∈ M, is associated a pure quantum state of the system: |ψ(λ)⟩, which is a normalized vector in a complex Hilbert space H. Since quantum states are defined up to a global phase, then the physical states are given by the identification: |ψ′(λ)⟩ = e−iα(λ)|ψ(λ)⟩. (1.6) This gauge freedom allows the introduction of the Berry connection, which captures how the quantum states changes when the parameters of the system are varied. If we consider a basis of normalized states |ψ(λ)⟩, the Berry connection is defined as the one-form βi = i⟨ψ(λ)|∂iψ(λ)⟩, (1.7) where ∂i = ∂ ∂λi , such that β = βidλ i take real values. Moreover, this connection defines a notion of parallel transport in parameter space [19]. From this connection, we can define the Berry curvature F , as the exterior derivative of this conection: F = dβ (1.8) with local components Fij = ∂iβj − ∂jβi. (1.9) The Berry curvature captures the geometric structure of the parameter space and plays a central role in understanding geometric phases and topological properties of quantum systems, as has been mentioned above. The Berrry conection plays a key role in the construction of the quantum geometric tensor, not only because it defines the Berry, but also because it is responsible for making the quantum metric tensor gauge invariant (1.6) . The QGT can be constructed from the derivatives of the normalized states |ψ(λ)⟩ with respect to the parameters of the system, giving it a positive-definite Hermitian structure over the parameter manifold. It is defined as [1] Qij = ⟨∂iψ|∂jψi⟩ − ⟨∂iψ|ψ⟩⟨ψ|∂jψi⟩ (1.10) As noted, the QGT is Hermitian: Q = Q† and can be decomposed into its symmetric (real) and antisymmetric (imaginary) parts: Real part — Quantum Metric Tensor: Gij = Re [Qij ] (1.11) Imaginary part — Berry curvature: −1 2 Fij = Im [Qij ] (1.12) The physical interpretation of the quantum geometric tensor (QGT) is that it encapsulates two funda- mental aspects of a quantum system: • The Quantum Metric Tensor (QMT): This tensor measures the distance between quantum states that are infinitesimally close in the parameter space. The QMT is related to the quantum fidelity and the quantum susceptibility3 and plays a crucial role in understanding quantum phase transitions. • The Berry curvature: This describes the topological non-triviality of the system. It can be thought as the geometric analogue of the electromagnetic field and is related to the geometric phases that arise after an adiabatic variation of the system’s parameters over a cycle. Thus, the QGT contains the fundamental aspects of the geometric structures that governs the dynamics of the system. In the following section we will derive the QMT from the overlap between two nearby states. 3In fact, the QMT and the quantum susceptibility are the same. 12 1.1.2 Quantum Metric Tensor. We want to construct a sense of a distance in quantum mechanics4. Following [1], let’s consider a family {ψ(λ)} of normalized vectors of an m-dimensional Hilbert space that depend smoothly on an n-dimensional parameter λ = (λ1, ..., λn) ∈ R n embedded in an n-dimensional manifold M. Then, taking advantage of the inner product defined over the Hilbert space, we have that the distance between two “close” states with respect to the parameters is given by d(ψ(λ+ δλ), ψ(λ)) = ∥ψ(λ+ δλ) − ψ(λ)∥, (1.13) which induces a metric in the following manner: ∥ψ(λ+ δλ) − ψ(λ)∥2 = ⟨ψ(λ+ δλ) − ψ(λ)|ψ(λ+ δλ) − ψ(λ)⟩ (1.14) up to second order ∥ψ(λ+ δλ) − ψ(λ)∥2 ≃ ⟨∂iψ(λ)|∂jψ(λ)⟩dλidλj (1.15) where the partial derivatives are taken with respect to the parameter ∂iψ(λ) = ∂ψ(λ) ∂λi (1.16) and the inner product is defined by ⟨ϕ(λ)|ψ(λ)⟩ = ∫ V ol dmx ϕ∗(λ)ψ(λ). (1.17) Due to the fact, that the inner product is a complex number we can consider its real and imaginary parts ⟨∂iψ(λ)|∂jψ(λ)⟩ = γij + iξij (1.18) which satisfy γij(λ) = γji(λ) & ξij(λ) = −ξji(λ). (1.19) Then, ∥ψ(λ+ δλ) − ψ(λ)∥2 ≃ γij(λ)dλidλj (1.20) Note that γij are the components of the quantum metric tensor over the manifold of collective states. This tensor is not well-defined over the manifold of physical states since it is not invariant under the gauge transformation: ψ(λ) → ψ′(λ) = eiα(λ)ψ(λ). (1.21) As we know, these two states define the same physical state. This means, that the quantum metric tensor must be invariant under this gauge transformation. Considering this transformation we have γ′ij = γij + βi(∂jα) + βj(∂iα) + (∂iα)(∂jα) (1.22) where βi(λ) are the components of the Berry connection defined as βi(λ) = −i⟨ψ(λ)|∂iψ(λ)⟩ (1.23) which under the gauge transformation βi → β′ i = βi + ∂iα. (1.24) Taking into account the normalization condition ⟨ψ(λ)|ψ(λ)⟩ = 1 (1.25) then, ∂i⟨ψ(λ)|ψ(λ)⟩ = ⟨∂iψ(λ)|ψ(λ)⟩ + ⟨ψ(λ)|∂iψ(λ)⟩ = 0 (1.26) 4In general, we are only interested in the transition amplitude between two quantum states without considering the relative distance between them. 13 making the Berry connection, β(λ), a real quantity. This ensures that the Berry curvature tensor is auto- matically gauge invariant under such transformations. Now, we define the quantum metric tensor as Gij(λ) ≡ γij(λ) + βi(λ)βj(λ) (1.27) which transforms as a tensor under coordinate transformations and is gauge invariant under the transforma- tion (1.21). Moreover, this tensor is symmetric and positive defined. Finally, we can obtain this same quantum metric tensor from the squared of the Fubini-Study distance between two rays, ψ̄(λ1) and ψ̄(λ2) in the projective Hilbert space associated to the normalized vectors ψ(λ1) and ψ(λ2) of the Hilbert space, respectively: DFS(ψ̄1, ψ̄2) = inf α1,α2 ∥ψ1e iα1 − ψ2e iα2∥ = 2 − 2∥⟨ψ1|ψ2⟩∥. (1.28) This framework sets the stage for the next chapter, where we extend these concepts to quantum systems defined on curved spaces, in which the metric depends explicitly on the parameters of the system. We will illustrate these ideas through concrete examples in one and two dimensions, showing how the geometry of the configuration space affects the quantum geometric tensor. 14 Chapter 2 Quantum Geometric Tensor: Parameter-dependent Spacetime metric (“Curved spacetime”). We are interested when the spacetime metric of the configuration space depends explicitly on the parameters of the system (λ), gµν = gµν(x, λ). Then, the inner product must be modified in a manner takes this into account: ⟨ϕ(λ)|ψ(λ)⟩ = ∫ V ol dNx √ gϕ∗(λ)ψ(λ) (2.1) where g is the determinant of the spacetime metric. This subtle change in the inner product induces an important change with respect to the normalization condition ⟨ψ(λ)|ψ(λ)⟩ = 1, in order to ∂ρ⟨ψ|ψ⟩ = ⟨∂ρψ|ψ⟩ + ⟨ψ|∂ρψ⟩ − 1 2 ⟨σρ⟩ = 0. (2.2) where we defined the deformation vector σρ to be σρ ≡ gµν∂ρg µν . (2.3) This object emerged purely because of the geometry of the system and is the responsible for the extra terms that appear in the generalization of the QGT. The derivative is with respect to the parameters ∂ρ = ∂ ∂λρ ρ = 1, ..., n. (2.4) 2.1 Quantum Metric Tensor We need to answer the question on how to define a distance in a parameter-dependent metric in a curved background. To do this, first, we need to take into account that the metric depends on the parameters λ, so when we vary with respect to λ the metric also is taken into account. Then the braket in the inner product must be thought as ⟨ϕ(λ)|ψ(λ)⟩ = ∫ V ol dNx ( g1/4(λ)ϕ(λ) )∗ ( g1/4(λ)ψ(λ) ) = ⟨g1/4(λ)ϕ(λ)|g1/4(λ)ψ(λ)⟩ (2.5) 15 where √ g was separated into two factors of g1/4. This is so to consider the variation of the metric that corresponds to the state where the parameter has been shifted infinitesimally. It is easy to note that the volume element of this inner product is invariant under coordinate transformations x→ x′ = f(x), (2.6) since the parameters λ do not depend on the coordinate x. Then, the metric transforms under the coordinate transformation as usual gµν(x′, λ) = ∂xα ∂x′µ ∂xβ ∂x′ν gαβ(x, λ) gµν(x′, λ′) = ∂xα ∂x′µ ∂xβ ∂x′ν gαβ(x, λ′). (2.7) In this manner, the Jacobian of dNx′ → dNx will be compensated with the determinant of the transformed metric. Now, under the assumption of the adiabatic approximation, we need to write the definition of distance between two infinitesimally separated states in parameter space, ψ(λ + δλ) and ψ(λ), in terms of the inner product thought as in (2.5): ∥ψ(λ+ δλ) − ψ(λ)∥2 = ⟨ψ(λ+ δλ) − ψ(λ)|ψ(λ+ δλ) − ψ(λ)⟩ = 2 − ⟨g1/4(λ+ δλψ(λ+ δλ)|g1/4(λ)ψ(λ)⟩ − ⟨g1/4(λ)ψ(λ)|g1/4(λ+ δλψ(λ+ δλ)⟩. (2.8) Up to second order, this equation takes the following form: ∥g1/4(λ+ δλ)ψ(λ+ δλ) − g1/4(λ)ψ(λ)∥2 = γρκδλ ρδλκ (2.9) where γρκ ≡ 1 2 ( ⟨g1/4∂ρψ|g1/4∂κψ⟩ + ⟨g1/4∂κψ|g1/4∂ρψ⟩ ) − 1 8 ( ⟨g1/4ψ|σκ|g1/4∂ρψ⟩ + ⟨g1/4ψ|σρ|g1/4∂κψ⟩ ) − 1 8 ( ⟨g1/4∂ρψ|σκ|g1/4ψ⟩ + ⟨g1/4∂κψ|σρ|g1/4ψ⟩ ) + 1 16 ⟨σρσκ⟩ (2.10) and ⟨σρσκ⟩ is the expectation value of σρσκ: ⟨σρσκ⟩ ≡ ⟨g1/4ψ|σρσκ|g1/4ψ⟩. (2.11) Again, this tensor is not gauge invariant (1.21) because the term ⟨g1/4∂ρψ|g1/4∂κψ⟩ + ⟨g1/4∂κψ|g1/4∂ρψ⟩ fails to be. To make this tensor invariant we define from (2.2) a modified Berry connection: βρ ≡ −i⟨g1/4ψ|g1/4∂ρψ⟩ + i 4 ⟨σρ⟩ (2.12) which is real and transforms under the gauge transformation as βρ → βρ + ∂ρα (2.13) since ⟨σρ⟩ is gauge invariant. 16 Then, the term βρβκ =⟨g1/4∂ρψ|g1/4ψ⟩⟨g1/4ψ|g1/4∂κψ⟩ − 1 4 ⟨σρ⟩⟨g1/4ψ|g1/4∂κψ⟩ − 1 4 ⟨σκ⟩⟨g1/4∂ρψ|g1/4ψ⟩ − 1 16 ⟨σρ⟩⟨σκ⟩ (2.14) will compensate the non-gauge invariance. We define the Quantum Metric Tensor (QMT) as Gρκ = γρκ − βρβκ. (2.15) which is symmetric and gauge-invariant. Explictly: Gρκ = 1 2 ( ⟨g1/4∂ρψ|g1/4∂κψ⟩ + ⟨g1/4∂κψ|g1/4∂ρψ⟩ ) − 1 8 ( ⟨g1/4ψ|σκ|g1/4∂ρψ⟩ + ⟨g1/4ψ|σρ|g1/4∂κψ⟩ ) − 1 8 ( ⟨g1/4∂ρψ|σκ|g1/4ψ⟩ + ⟨g1/4∂κψ|σρ|g1/4ψ⟩ ) − 1 2 ( ⟨g1/4∂ρψ|ψ⟩⟨g1/4ψ|g1/4∂κψ⟩ + ⟨g1/4∂κψ|g1/4ψ⟩⟨g1/4ψ|g1/4∂ρψ⟩ ) + 1 8 ( ⟨σρ⟩⟨g1/4ψ|g1/4∂κψ⟩ + ⟨σκ⟩⟨g1/4ψ|g1/4∂ρψ⟩ ) + 1 8 ( ⟨σρ⟩⟨g1/4∂κψ|g1/4ψ⟩ + ⟨σκ⟩⟨g1/4∂ρψ|g1/4ψ⟩ ) + 1 16 ⟨σρσκ⟩ − 1 16 ⟨σρ⟩⟨σκ⟩. (2.16) where all the additional terms to the usual case becomes from the deformation vector σρ as we have mentioned. 2.1.1 QMT: Quantum Fidelity and Quantum Susceptibility. One of the most interesting quantities in quantum mechanics is the overlap between two quantum states, which we will consider pure states. This overlap is the transition amplitude between the states, while in quantum information measures how close are these two states. Moreover, the overlap measures the loss of information in quantum transportation between one state to other for a long distance. Thus, we define the overlap between two pures states as f(ψ′, ψ) = ⟨ψ′|ψ⟩ (2.17) and the fidelity is defined as the modulus of the overlap: F (ψ′, ψ) = |⟨ψ′|ψ⟩|. (2.18) Now, let’s consider that our quantum states depend adiabatically on one external n-dimensional parameter λ and are very close to each other: ψ′(λ) = ψ(λ + δλ). Also, we must consider that the spacetime metric depends on this parameter, then the fidelity is F (ψ′, ψ) = ∥⟨(g1/4ψ)′|g1/4ψ)⟩∥. (2.19) Changing into a more specific notation, the above equation can be written as F (λ+ δλ, λ) = ∥⟨g1/4(λ+ δλ)ψ(λ+ δλ)|g1/4(λ)ψ(λ)⟩∥ (2.20) so we emphasize that we are considering two close states in the parameter manifold. 17 To calculate the fidelity is easier to start with the squared of it: |f(λ+ δλ, δλ)|2 = ⟨(g1/4ψ)′|g1/4ψ⟩⟨g1/4ψ|(g1/4ψ)′⟩. (2.21) Then, up to second order we get |f(λ+ δλ)|2 =1 + δλρ ( ⟨g1/4∂ρψ|g1/4ψ⟩ + ⟨g1/4ψ|g1/4∂ρψ⟩ − 1 2 ⟨σρ⟩ ) + δλρδλκ 2 ( ⟨g1/4∂ρ∂κψ|ψ⟩ + ⟨g1/4ψ|g1/4∂ρ∂κψ⟩ − 1 2 ⟨g1/4∂κψ|σρ|g1/4ψ⟩ − 1 2 ⟨g1/4ψ|σρ|g1/4∂κψ⟩ − 1 2 ⟨∂κσρ⟩ + 1 8 ⟨σρσκ⟩ ) + δλρδλκ ( ⟨g1/4∂ρψ|g1/4ψ⟩⟨g1/4ψ|g1/4∂κψ⟩ − 1 4 ⟨σρ⟩⟨g1/4ψ|g1/4∂κψ⟩ − 1 4 ⟨σκ⟩⟨g1/4∂ρψ|g1/4ψ⟩ + 1 16 ⟨σρ⟩⟨σκ⟩ ) (2.22) then, using that ∂κ∂ρ⟨g1/4ψ|g1/4ψ⟩ = ⟨g1/4∂κ∂ρψ|g1/4ψ⟩ + ⟨g1/4ψ|g1/4∂κ∂ρψ⟩ + ⟨g1/4∂ρψ|g1/4∂κψ⟩ + ⟨g1/4∂κψ|g1/4∂ρψ⟩ − 1 2 ⟨g1/4∂ρψ|σκ|g1/4ψ⟩ − 1 2 ⟨g1/4∂κψ|σρ|g1/4ψ⟩ − 1 2 ⟨g1/4ψ|σκ|g1/4∂ρψ⟩ − 1 2 ⟨g1/4ψ|σρ|g1/4∂κψ⟩ − 1 2 ⟨∂κσρ⟩ + 1 4 ⟨σρσκ⟩ = 0 (2.23) we can rewrite (2.22) as |f(λ+ δλ)|2 =1 − δλρδλκ [1 2 ( ⟨g1/4∂ρψ|g1/4∂κψ⟩ + ⟨g1/4∂κψ|g1/4∂ρψ⟩ ) − ( ⟨g1/4∂ρψ|g1/4ψ⟩⟨g1/4ψ|g1/4∂κψ⟩ ) − 1 8 ( ⟨g1/4∂ρψ|σκ|g1/4ψ⟩ + ⟨g1/4∂κψ|σρ|g1/4ψ⟩ ) − 1 8 ( ⟨g1/4ψ|σκ|g1/4∂ρψ⟩ + ⟨g1/4ψ|σρ|g1/4∂κψ⟩ ) + 1 8 ( ⟨σρ⟩⟨g1/4ψ|g1/4∂κψ⟩ + ⟨σκ⟩⟨g1/4ψ|g1/4∂ρψ⟩ ) + 1 8 ( ⟨σκ⟩⟨g1/4∂ρψ|g1/4ψ⟩ + ⟨σρ⟩⟨g1/4∂κψ|g1/4ψ⟩ ) + 1 16 ⟨σρσκ⟩ − 1 16 ⟨σρ⟩⟨σκ⟩ ] . (2.24) where we used ∂κ∂ρ⟨g1/4ψ|g1/4ψ⟩ = 0 and that that the linear term in (2.24) is zero due to the normalization condition (2.2). We take the square root of (2.24) to obtain up to second order the fidelity: F (λ+ δλ, λ) = 1 − δλρδλκ 2 χρκ (2.25) 18 where χρκ is the quantum susceptibility given by χρκ ≡1 2 ( ⟨g1/4∂ρψ|g1/4∂κψ⟩ + ⟨g1/4∂κψ|g1/4∂ρψ⟩ ) − 1 8 ( ⟨g1/4∂ρψ|σκ|g1/4ψ⟩ + ⟨g1/4∂κψ|σρ|g1/4ψ⟩ ) + − 1 8 ( ⟨g1/4ψ|σκ|g1/4∂ρψ⟩ + ⟨g1/4ψ|σρ|g1/4∂κψ⟩ ) + − ⟨g1/4∂ρψ|g1/4ψ⟩⟨g1/4ψ|g1/4∂κψ⟩ + 1 4 ⟨σρ⟩⟨g1/4ψ|g1/4∂κψ⟩ + 1 4 ⟨σκ⟩⟨g1/4∂ρψ|g1/4ψ⟩ + 1 16 ⟨σρσκ⟩ − 1 16 ⟨σρ⟩⟨σκ⟩ (2.26) Thus, the quantum susceptibility χρκ = γρκ − βρβκ (2.27) is the same as the quantum metric tensor. Where γρκ and βρ are defined as (2.10) and (2.12) respectively. 2.2 Berry Curvature. To see how from the normalization condition (2.2) we defined the Berry conection (2.12) we should note that 0 = ∂ρ⟨g1/4ψ|g1/4ψ⟩ = ⟨g1/4∂ρψ|g1/4ψ⟩ + ⟨g1/4ψ|g1/4∂ρψ⟩ − 1 2 ⟨g1/4ψ|σρ|g1/4ψ⟩ (2.28) thus, we define the modified Berry connection as βρ = −i⟨ψ|∂ρψ⟩ + i 4 ⟨σρ⟩ (2.29) Properties of the Berry connection: • It is real (same as the usual case of Provost-Vallee [1]). • It transforms as a connection in the parameter manifold. To show that the Berry connection is real we should remember that the spacetime metric is a real (symmetric) tensor, therefore, σρ is real too. Let βρ denote the complex conjugate of βρ, then βρ = i ⟨ψ|∂ρψ⟩ − i 4 ⟨σρ⟩ = i⟨∂ρψ|ψ⟩ − i 4 ⟨σρ⟩ (2.30) and after multiplying (2.28) by i we get that −i⟨ψ|∂ρψ⟩ + i 4 ⟨σρ⟩ = i⟨∂ρψ|ψ⟩ − i 4 ⟨σρ⟩ (2.31) namely, βρ = βρ (2.32) which, is the reality condition. 19 To check that this Berry connection transforms as a connection in the parameter space, let us consider a coordinate transformation given by1 λ→ λ′ (2.33) and let us consider that ψ′(λ′) = ψ(λ) and g′(λ′) = g(λ). Note that σ′ ρ(λ ′) = g′µν(λ′)∂′ρg ′µν(λ′) = gµν(λ) ∂λκ ∂λ′ρ ∂κg µν = ∂λκ ∂λ′ρ σκ (2.34) Then, β′ ρ = −i⟨g′1/4(λ′)ψ′(λ′)|g′1/4(λ′) ∂′ρψ ′(λ′)⟩ + i 4 ⟨g′1/4(λ′)|σ′ ρ(λ ′)|g′1/4(λ′)ψ′(λ′)⟩ = −i⟨g1/4(λ)ψ(λ)|g1/4(λ) ∂λκ ∂λ′ρ ∂κψ(λ)⟩ + i 4 ⟨g1/4(λ)ψ(λ)| ∂λ κ ∂λ′ρ σκ|g1/4(λ)ψ(λ)⟩ = −i ∂λ κ ∂λ′ρ ⟨g1/4ψ|g1/4∂κψ⟩ + i 4 ∂λκ ∂λ′ρ ⟨σκ⟩ (2.35) Therefore, β′ ρ = ∂λκ ∂λ′ρ βκ (2.36) proving that β is a one form in the parameter space, transforming as a connection as expected under phase transformations. Note that we are able to take out the ∂λκ ∂λ′ρ term because the inner product is defined over the space coordinates x, i.e., we do not integrate over the parameters λ. Now, following that by definition the Berry curvature is the exterior derivative of the Berry connection F = dβ (2.37) then, the components are Fρκ = ∂ρβκ − ∂κβρ = −i ( ⟨∂ρψ|∂κψ⟩ − ⟨∂κψ|∂ρψ⟩ ) + i 4 ( ⟨ψ|σρ|∂κψ⟩ − ⟨ψ|σκ|∂ρψ⟩ ) + i 4 ( ⟨∂ρψ|σκ|ψ⟩ − ⟨∂κψ|σρ|ψ⟩ ) (2.38) where we have omitted the g1/4 factors inside the brakets. 2.3 Quantum Geometric Tensor Curved Background. We just need one more ingredient to be able to define the Quantum Geometric Tensor (QGT) in a curved background. Let’s consider the following projection operator P = I− |g1/4ψ⟩⟨g1/4ψ|. (2.39) It’s easy to verify that P 2 = P. P 2 = ( I− |g1/4ψ⟩⟨g1/4ψ| )( I− |g1/4ψ⟩⟨g1/4ψ| ) = I− |g1/4ψ⟩⟨g1/4ψ| − |g1/4ψ⟩⟨g1/4ψ| + |g1/4ψ⟩⟨g1/4ψ|g1/4ψ⟩⟨g1/4ψ| = I− |g1/4ψ⟩⟨g1/4ψ| − |g1/4ψ⟩⟨g1/4ψ| + |g1/4ψ⟩⟨g1/4ψ| = P (2.40) 1This coordinate transformation is independent of the configuration space coordinates x. 20 where the normalization condition ⟨g1/4ψ|g1/4ψ⟩ = 1 was used. Now, we define the Quantum Geometric Tensor in curved background as Qρκ ≡ ⟨∂ρ ( g1/4ψ ) |P|∂κ ( g1/4ψ ) ⟩ (2.41) We write explicitly the terms of the QGT omitting the g1/4 factors: Qρκ = ⟨∂ρψ|∂κψ⟩ − ⟨∂ρψ|ψ⟩⟨ψ|∂κψ⟩ − 1 4 ⟨ψ|σρ|∂κψ⟩ − 1 4 ⟨∂ρψ|σκ|ψ⟩ + 1 4 ⟨σρ⟩⟨ψ|∂κψ⟩ + 1 4 ⟨σκ⟩⟨∂ρψ|ψ⟩ + 1 16 ⟨σρσκ⟩ − 1 16 ⟨σρ⟩⟨σκ⟩. (2.42) where the deformation vector is responsible for all the extra terms, which are purely geometric. The relevance of this tensor is that it contains the fundamental structures of the parameter space: The quantum metric tensor and the Berry curvature. This quantities are given by taking the real (symmetric) part: Re (Qρκ) = Gρκ (2.43) and the imaginary (anti-symmetric) part: Im (Qρκ) = −1 2 Fρκ (2.44) 2.4 Examples of QGT in curved background. Let us consider a Lagrangian of the form: L = 1 2 gij(x, λ)ẋiẋj − V (x, λ), (2.45) that corresponds to a particle moving in a curved space with metric gij(x, λ), which depends on some parameter λ ∈ M. Then, the Hamiltonian is given by H = 1 2 gijpipj + V (x, λ). (2.46) The indices i, j = 1, ..., N , where n is the dimension of the configuration space (the physical space). To quantize the system and construct the Schrödinger equation in the coordinate representation we need to introduce the Laplace-Beltrami operator [20]. gijpipj → ∇2ψ = 1√ g ∂ ∂xj (√ ggij ∂ψ ∂xi ) . (2.47) In this way, the whole gijpipj corresponds to a Hermitian operator under the inner product (2.1). It will be useful to decompose the metric into the “inverse” of the vielbeins eia(x, λ) as follows: gij = eiae aj , s.t. ηab = eaiebi (2.48) where ηab is an N-dimensional flat space metric and ebj is the tetrad or vierbein. Thus, the Laplace-Beltrami operator is given by ∇2ψ = 1 e ∂ ∂xj ( e eia e aj ∂ψ ∂xi ) (2.49) where e is the determinant of the vierbein eai . 21 2.4.1 One-dimensional Curved Anharmonic Oscillator. The Lagrangian for the one-dimensional curved anharmonic oscillator that we are going to study is given by L = 2λx2ẋ2 − ω2 2 λx4, (2.50) which is easy to note that the metric is g = 4λx2. (2.51) The Hamiltonian of the system is H = 1 8 p2x λx2 + ω2 2 λx4. (2.52) Using the Laplace-Beltrami operator (2.47) we can quantize the system and obtain the time-independent Schrödinger equation: ( − ℏ 2 8λx2 d2 dx2 + ℏ 2 8λx3 d dx + ω2 2 λx4 ) ψn(x) = Enψn(x) (2.53) with solutions given by ψn(x) = 1√ 2nn! ( ω πℏ )1/4 e− ωλx4 2ℏ Hn ( √ ωλ ℏ x2 ) (2.54) for n = 0, 1, 2, ..., and Hn(x) are the Hermite polynomials. The energy eigenvalues are the same as for the harmonic oscillator: En = ℏω ( n+ 1 2 ) , n = 0, 1, 2, ... (2.55) The solutions of the system doesn’t have an imaginary part. This tells us that there is no Berry curvature, so the quantum geometric tensor is equal to the quantum metric tensor. Now, we compute the deformation vector σρ for ρ ∈ {λ, ω}: σλ = − 1 λ , (2.56) σω = 0. (2.57) For this system we were able to compute the QGT for the n-excited state:2 G[n] = (n2 + n+ 1) ( 1 8λ2 1 8λω 1 8λω 1 8ω2 ) . (2.58) This metric tensor is singular, i.e., its determinant equals zero. The reason of this is that the parameters λ and ω are not independent of each other. Nevertheless, we can still get some meaningful information by taking a closer look to the QGT components. We observe that as either λ or ω approach zero, the components of the QGT diverge. This divergence signals the presence of a possible quantum phase transition of the system. Specifically, when ω → 0, the system reduces to that of a free particle, while in the limit λ → 0, the Hamiltonian becomes unbounded and the Lagrangian vanishes. 2.4.2 Oscillator with a Morse type potential. Let’s consider a potential that corresponds to the short-range repulsion term of the Morse potential in one- dimensional curved space. Then, the Lagrangian of the system is L = λ2 8 e−λx ẋ2 − ω2 2 e−λx (2.59) with Hamiltonian H = 2 λ2 eλxp2x + ω2 2 e−λx. (2.60) 2The dependence of the quantum number n is given between brackets to avoid confusion with the components of the QGT. 22 It’s easy to see that the metric of the system is g = λ2 4 e−λx (2.61) which depends explicitly on x and on the parameter λ. Now, we show the deformation vector, σρ with ρ ∈ {λ, ω} σλ = x− 2 λ , (2.62) σω = 0. (2.63) Before computing the QGT, let’s analyse the phase space for this system: In Figure 2.1. In figure (a) it is shown the phase diagram for different energies, in figure (b) for different λ, in (c) for different ω and in (d) the interesting symmetry that appears when changing the value λ → −λ. Interestingly, the loop gets “bigger” and wider for increasing energy while by increasing ω is the other way around. For increasing λ, we can see that the loop gets wider but approaches 0 in the x axis. Moreover, it presents the interesting fact that the loop gets inverted symmetrically by changing the sign of the value of λ. We observe also that the system has a singularity at x→ ∞ for λ > 0 and at x→ −∞ for λ < 0. Figure 2.1: Phase diagrams for the Morse-like potential. (a) H(ω, λ) = Ei. (b) H(ω, λi) = 1 (c) H(ωi, λ) = 1 (d) H(ω,±λi) = 1 23 Again, the time-independent Schrödinger equation is obtained from the Laplace-Beltrami operator (2.47): [ − 2ℏ2 λ2 eλx ( λ 2 d dx + d2 dx2 ) + ω2 2 e−λx ] ψn(x) = Enψn(x). (2.64) For this system, we will only be able to compute the QGT for the ground-state ψ0(x), which is given by ψ0(x) = Ae− ω 2ℏ e−λx . (2.65) with energy eigenvalue E0 = ℏω 2 . (2.66) It’s important to compute the normalization constant A because this depends on the parameters and it is crucial for the computation of the QGT. So, using the normalization condition ⟨ψ|ψ⟩ = 1 with the inner product of the curved space we have ⟨ψ0|ψ0⟩ = A2 ∞ ∫ −∞ dx ( λ 2 e− λ 2 xe− ω ℏ e−λx ) . (2.67) If we perform the change of variable u = e− λ 2 x and noticing that u x→−∞−−−−−→ 0 and u x→∞−−−−→ −∞ we arrive to ⟨ψ0|ψ0⟩ = A2 ∞ ∫ 0 e− ω ℏ u2 du (2.68) which is the usual harmonic oscillator constrained to the positive real line R +. Thus the normalization constant A is A = √ 2 ( ω πℏ ) 1 4 (2.69) We need to point out that the normalization constant differs from usual the harmonic oscillator by a factor of √ 2. Analogously as in the previous example, ψ0(x) doesn’t have an imaginary part, thus, the Berry connection is, again, zero and the QGT is the same as the QMT. The components of the QGT for the ground-state are: Gλλ =⟨∂λψ|∂λψ⟩ − ⟨∂λψ|ψ⟩⟨ψ|∂λψ⟩ + 1 2 ⟨σλ⟩⟨∂λψ|ψ⟩ − 1 2 ⟨∂λψ|σλ|ψ⟩ + 1 16 ⟨σ2 λ⟩ − 1 16 ⟨σλ⟩2 (2.70) Gλω = ⟨∂λψ|∂ωψ⟩ − 1 4 ⟨∂ωψ|σλ|ψ⟩ (2.71) Gωω = ⟨∂ωψ|∂ωψ⟩ (2.72) For completion, we write the components for the QGT for the ground state explicitly: Gλλ = 1 16λ2 [ 4 + 2(γ − 4)γ + π2 + 2 ln(4)2 + 4(γ − 2) ln ( 4ω ℏ ) + +2 ln (ω ℏ ) ln ( 16ω ℏ )] (2.73) Gλω = 1 16λω { 2 − 2γ + 2erf ( √ ω ℏ ) + ln ( ℏ 2 16ω2 ) + 1√ π [ 2 G3,0 2,3 ( ω ℏ ∣ ∣ ∣ 1, 1 0, 0, 32 ) − G3,0 2,3 ( ω ℏ ∣ ∣ ∣ 1, 1 0, 0, 12 )]} (2.74) 24 Gωω = 1 8ω2 (2.75) where γ is the Euler constant, erf(z) the error function and Gm,n p,q ( z ∣ ∣ ∣ a1, ..., an, an+1, ..., ap b1, ..., bm, bm+1, ..., bq ) the MeijerG function. In figure 2.2 we show the graphics for the components of the QGT, where we can note that the components of Gλλ and Gωω are positive (graphics (2.2a) and (2.2b), respectively), while the component Gλω (2.2c) presents a change of sign. In the contour plot (2.2d) we plot Gλω for different values of λ, it is shown that the ω-axis is cut in the same point given by ω0 ∼ 1.03716 and in (2.2e) we plot Gλω for different values of ω and we can appreciate that in the critical point ω = ω0 the curves change sign too. This critical point ω0 ∼ 1.03716, most appreciated in figure (2.2d), seems to have an impact in the behavior of the system, thus it can be related to figure (2.1c) where it makes softer the so abrupt increase and decrease of momentum and accelerating back to infinity for ω < ω0 where Gλω is negative. More so, the QGT presents the particularity of being non-singular; that is, it has a determinant different from zero, in contrast with the QGT of the harmonic oscillator. Furthermore, all the components of the QGT show a quantum phase transition for ω = 0 or λ = 0 where the system changes to a free particle. This transition is in some sense equivalent to the one presented in the phase space in the limit x → ±∞, nevertheless here is observed for any velocity, whereas in the phase space exists only for null velocity. 2.4.3 Generalized Harmonic Oscillator in curved space. Up until now, all our systems have null Berry curvature. Thus, we base the following example on the generalized harmonic oscillator, which from the literature [12, 16, 21] is a well-known system with Berry curvature different from zero, where the Hamiltonian is H = 1 2 [ cx2 + b(xp+ px) + ap2 ] (2.76) and by a straightforward calculation we get the Lagrangian: L = 1 2a ẋ2 − Ω 2 x2 − b 2a (xẋ+ ẋx) (2.77) with Ω = c− b2 a . (2.78) We will consider the generalized anharmonic oscillator with a = 1 and spatial metric:3 g = 4λx2 (2.79) and Hamiltonian H = ap2 8λx2 + b 4 (xp+ px) + cλx4 2 (2.80) so that the Lagrangian is L = 2λx2ẋ2 a − bλ a (x3ẋ+ ẋx3) − Ω 2 λx4 (2.81) In this case the time-independent Schrödinger equation is given by ( − ℏ 2 8λ ( 1 x2 ∂2x − 1 x3 ∂x ) − iℏbx 2 ∂x − iℏb 2 + cλx4 2 ) ψn(x) = Enψ(x) (2.82) where the time-independent solutions are ψn(x) = ( ω πℏ )1/4 1√ 2nn! e− ωλx4 2ℏ Hn ( √ ωλ ℏ x2 ) e− ibλx4 2ℏ (2.83) 3We choose a = 1, in order to avoid that the QGT of the generalized harmonic oscillator to be singular. 25 Figure 2.2: Components of the QGT for the Morse-like potential. (a) Gλλ (b) Gωω (c) Gλω (d) Contour Plot of Gωλ, λ = (0.01, 0.05,−0.01,−0.05). (e) Contour Plot of Gωλ, ω = (1.0371, 1.03714, 1.03718, 1.0372). and the energy eigenvalues are the same as the generalized harmonic oscillator: En = ( n+ 1 2 ) ℏω (2.84) where ω = √ c− b2. We only miss the deformation vector σρ, ρ ∈ {λ, ω} σλ = − 1 λ , (2.85) σω = 0, (2.86) 26 Figure 2.3: Determinant of the QGT for the oscillator with Morse type potential. (a) Determinant of QMT (b) Contour plot of the determinant with ω = c. (c) Contour plot of the determinant with λ = c. These contour plots of the determinant show that when increasing the absolute value of the parameter λ or ω the determinant tends to zero, but it’s always positive. In (a) we are leaving ω constant, while in (b) we set λ fixed. 27 which are the same as for the anharmonic oscillator. Then, the QMT for the n-excited state is G[n] = (n2 + n+ 1)   c 8ω2λ2 0 1 16ω2λ 0 c 8ω4 − b 16ω4 1 16ω2λ − b 16ω4 1 32ω4   (2.87) which is degenerated. In this case, the Berry curvature is different from zero. In fact, using equation (2.38) we have that for the n-excited state it is given by Fρκ[n] = 2n+ 1 16ω3λ   0 2c −b −2c 0 −λ b λ 0   . (2.88) This curvature reduces to the Berry curvature of the standard anharmonic oscillator, as given in [16], in the limit λ→ ∞. 2.4.4 Coupled Anharmonic Oscillator in curved space. Now, we present an example in two dimensions: The coupled anharmonic oscillator in curved space with spatial metric g = ( a2x2 0 0 b2y2 ) (2.89) which is a diagonal matrix dependent explicitly on the parameters a and b. Then the Lagrangian is L = 1 2 a2x2ẋ2 + 1 2 b2y2ẏ2 − k1 2 ( a2 4 x4 + b2 4 y4 ) − k2 2 ( a 2 x2 − b 2 y2 )2 (2.90) so that the Hamiltonian is H = p2x 2a2x2 + p2y 2b2y2 + k1 2 ( a2 4 x4 + b2 4 y4 ) + k2 2 ( a 2 x2 − b 2 y2 )2 . (2.91) Since we are interested in the QGT, we need to quantize this system. To do so, we will consider the Laplace- Beltrami operator (2.47): ∇2ψ = 1√ abxy ( ∂x ( by ax ∂ψ ∂x ) + ∂y ( ax by ∂ψ ∂y )) = 1 a2x2 ∂2ψ ∂x2 − 1 a2x3 ∂ψ ∂x + 1 b2y2 ∂2ψ ∂y2 − 1 b2y3 ∂ψ ∂y (2.92) Thus, the time-independent Schrödinger equation is ĤΨn(x, y) = − ℏ 2 2a2x2 ∂2Ψn(x, y) ∂x2 + ℏ 2 2a2x3 ∂Ψn(x, y) ∂x − ℏ 2 2b2y2 ∂2Ψn(x, y) ∂y2 + ℏ 2 2b2y3 ∂Ψn(x, y) ∂y + k1 2 ( a2x4 4 + b2y4 4 ) Ψn(x, y) + k2 2 ( ax2 2 − by2 2 )2 Ψn(x, y) = EnΨn(x, y) (2.93) The ground state solution is given by Ψ0(x, y) = Aexp [ −ω1a 2x4 8 − ω2b 2y4 8 − β abx2y2 4 ] (2.94) with ω1 = √ k1 = ω2, β = 1 2 ( √ k1 − √ k1 + 2k2) < 0 28 and A is the normalization constant which will be obtained later. The energy of this ground-state is E0 = 1 2 (ω+ + ω−) (2.95) where ω+ = √ k1, ω− = √ k1 + 2k2 are the frequencies of the normal modes. Then, we can write our ground-state solution as Ψ0(U+, U−) = Aexp [ −1 2 ( ω+U 2 + + ω−U 2 − ) ] (2.96) where we have defined U± = 1√ 2 ( ax2 2 ± by2 2 ) . (2.97) Now, it is time to compute the normalization constant so we can compute the QGT. Note that the inner product in this case is given by ⟨ψ|ϕ⟩ = ∞ ∫ −∞ ∞ ∫ −∞ dxdy √ a2b2x2y2ψ∗(x, y)ϕ(x, y) (2.98) Then for the ground-state: 4 ⟨Ψ0|Ψ0⟩ = ∞ ∫ 0 ∞ ∫ 0 dxdy 4ab xy A2exp [ −ω1a 2x4 4 − ω2b 2y4 4 − β abx2y2 2 ] = ∞ ∫ 0 U+ ∫ −U+ dU+dU− A2exp [ − ( ω+U 2 + + ω−U 2 − )] = 4A2arctan ( √ ω− ω+ ) √ ω+ω− = 1. (2.99) We need to note that the region of integration stopped to be the whole plane with the change of variables, instead one only integrates on the region shown in Figure 2.4, which is the upper cone delimited by U+ = |U−|. Then, the ground-state solution is given by Ψ0(x, y) = (k1(k1 + 2k2)) 1/8 2 √ arctan ( 1 + 2k2 k1 )1/4 exp [ − √ k1 8 a 2 x 4 − √ k1 8 b 2 y 4 − 1 2 ( √ k1 − √ k1 − 2k2) abx2y2 4 ] (2.100) where we have written explicitly the parameters of the system: {k1, k2, a, b}. Since the wave-function does not have an imaginary part the Berry curvature is zero, thus the QGT is the same as the QMT. Due to the fact that the algebraic expressions of the QMT don’t give any clear information of the behaviour of it, we show in figure (2.5) the plots or, more specific, the projections of the components of the QMT. First thing to notice is that the plots of Gbk1 and Gbk2 are not to be found. This is because if one interchanges the parameter a and b they are the same as Gak1 and Gak2 respectively, this has a huge impact in the behaviour of the QMT making it singular. Second thing to note is the dependence of the components of the QMT on the parameters of the system, since Gk1k1 , Gk2k2 and Gk1k2 only depend on the spring constant k1 and the coupling constant k2 while the rest of the components depend also on the parameter a and the component Gab depends on all four parameters k1, k2, a and b with the peculiarity that we can interchange a and b. Thirdly, the component for Gab with a = −1 and b = 1 is just a translation by 1 2 4To explain the change on the limits of integration and the factor of 4 we took into consideration the definition √ x2 = |x|. 29 Figure 2.4: Region of Integration of Gaa with a = 1. Finally, even though the plots for Gak2 with k1 = 1 and k2 = 1 look alike their difference is not a trivial function. It’s easy to observed that when k1 and a goes to zero, the components of the QMT goes to infinity, but it is not the case when k2 goes to zero. In fact the term Gk1k1 ∼ 1 k2 1 recovering the expression for the usual anharmonic oscillator. This is not surprising since k2 = 0 means that the system is decoupled. Now, the terms Gk1k2 and Gk2k2 also take the form of ∼ 1 k2 1 , Gaa ∼ 1 a2 , Gkia ∼ 1 ak1 when i = 1, 2, and Gab = 0. These results are not so unexpected since they mean that the QMT keeps some information of the dimension of the space of parameters, which is a purely quantum effect. As mentioned before, the determinant of the QMT is zero, which can be avoided by setting one of the parameters a or b to be constant. This enables us to plot the subdeterminant of the QMT in figure (2.6) where we denoted as a suffix the parameter b set as a constant. One can see that it is positive definite and it diverges when k1, or a approaches to zero. But, when k2 → 0, then this subdeterminant takes the form ∼ 1 a2k4 1 . Therefore, we can see how the QMT, more specific the subdeterminant DetQMTb, detects two different quantum phase transitions: a → 0 the system collapse into a one dimensional modified harmonic oscillator, and the more interesting case k1 → 0, where, at first glance, one could think it becomes the linear coupled harmonic oscillator [13, 22], but in reality the system becomes a coupled harmonic oscillator with only two parameters [23]. 30 Figure 2.5: Components of the QMT for the coupled anharmonic oscillator in a curved space. (a) Gk1k1 (b) Gk2k2 (c) Gk1k2 (d) Gaa, k1 = 1 (e) Gaa, k2 = 1 (f) Gaa, a = 1 (g) Gak1 , k1 = 1 (h) Gak1 , k2 = 1 (i) Gak1 , a = 1 (j) Gak2 , k1 = 1 (k) Gak2 , k2 = 1 (l) Gak2 , a = 1 (m) Gab, k1=1, k2=1 (n) Gab, k1=1, b=1 (o) Gab, k2=1, b=1 (p) Gab, a=1, b= 1 31 Figure 2.6: Subdeterminant of the QMT of the coupled anharmonic oscillator in a curved space. (a) DetQMTb(k1 = 1, k2, a) (b) DetQMTb(k1, k2 = 1, a) (c) DetQMTb(k1, k2, a = 1) 32 Chapter 3 QGT for spin-1 2 particles. In previous chapters, we have calculated the QGT for particles satisfying the Schrödinger equation without spin. Moreover, most of the literature is focused on the QGT in non-relativistic quantum mechanics. Thus, it is natural to ask about particles with spin and to extend the concept of the QGT for relativistic systems. The relativistic effects can be tackled by using the path integral approach [12] where the geometric structure of quantum states emerges from functional integration. However, in this work we will focus on the explicit formulation of the Dirac equation: (γµpµ +m)ψ(x) = 0, (3.1) where the γ matrices satisfy the Clifford algebra {γµ, γν} = 2ηµνI, µ, ν = 0, 1, 2, 3. (3.2) where η is the Minkowski metric with signature (+,−,−,−). This equation possesses several important features: • It is covariant and takes the relativistic effects into account. • Intrinsic spin naturally emerges from the Dirac equation. • In condensed matter physics, Dirac equation arises as an effective low-energy description of various systems (spin-orbit coupling [24], Quantum Hall effect [25,26], graphene [27,28], topological insulators [29, 30], Weyl semimetals [31, 32], and many more systems). • The QGT can be calculated for particles and anti-particles (for condensed matter systems: particles- holes). With this in mind, and noting that an explicit calculation of the QGT for the Dirac equation is rarely found in the literature, we will analyze the QGT for the free spin- 12 particle in two representations of the γ matrices: Dirac and Weyl. We further extend our analysis of the QGT to the simplest Weyl-semimetal and, unlike the conventional approach, which relies on the Berry curvature and Chern number, we obtain the Bloch sphere structure directly from the QGT. This provides a novel perspective on how quantum geometry encodes topological information. To this end, we need to establish the proper inner product for Dirac spinors. Analogously to the case of the Schrödinger equation, where the probability density ρ = ψ†ψ is a conserved quantity, we know that the Dirac equation is invariant under the global gauge transformation ψ(x) → eiαψ(x). (3.3) Therefore, by Noether’s theorem, there is a conserved charge and current. Introducing the 4-current density jµ = ψ̄γµψ, the continuity equation is1 ∂µj µ = 0, (3.4) 1Recall that ψ̄ = ψ†γ0. 33 which, for µ = 0 gives the probability density which is always positive definite: j0 = ρ = ψ†ψ (3.5) Thus, the inner product will be defined with respect to this conserved quantity. 3.1 QGT for Dirac free particle. Before moving on to the explicit calculation of the QGT for the free particle solution of the Dirac equation let us recall that it has the general form ψ(x) = N u(p) eipµx µ (3.6) where u(p) is a four-component spinor (to simplify notation we will write up). In what follows, we will absorb the normalization constant N into up, but in later sections it will be given explicitly since, in general, it is a function of the parameters. As mentioned, we will consider as parameters the mass, m, and one component of the spatial momentum, say pz. Note that the energy eigenvalues are constrained by the dispersion relation E± = ± √ p⃗ 2 +m2 (3.7) thus making p0 dependent on these parameters. Affirmation: The exponential factors do not contribute to the QGT in general; that is, for the free particle theory the QGT is given by: Qij = ⟨∂iup|∂jup⟩ − ⟨∂iup|up⟩⟨up|∂jup⟩ (3.8) To prove this, we compute the components of the QGT for the spin- 12 particle as follows: First, we will show the partial derivatives with respect to the parameters of the free particle solution: ∂mψ = (∂mup − it(∂mEp)up) e−ipµx µ (3.9) ∂mψ † = ( ∂mu † p + it(∂mEp)u† p ) eipµx µ (3.10) ∂pzψ = (∂pzup − it(∂pzEp)up − izup) e−ipµx µ (3.11) ∂pzψ † = ( ∂pzu † p + it(∂pzEp)u† p + izu† p ) eipµx µ (3.12) Secondly, we will compute a general expression representing the products that appear when calculating the QGT components: Qij = ⟨∂iψ|∂jψ⟩ − ⟨∂iψ|ψ⟩⟨ψ|∂ψ⟩ = ⟨∂i(upe−ipµx µ )|∂j(upe−ipµx µ )⟩ − ⟨∂i(upe−ipµx µ )|upe−ipµx µ⟩⟨upe−ipµx µ |∂j(upe−ipµx µ )⟩, (3.13) which can be separated into the following equations. (∂iu † p + iα u† p ) · (∂jup − iβ up) = ∂iu † p · ∂jup + i(αu† p · ∂jup − β∂iu † p · up) + αβ, (3.14) and (∂iu † p · up + iα)(u† p · ∂jup − iβ) = (∂iu † p · up)(u† p · ∂jup) + i(αu† p · ∂jup − β∂iu † p · up) + αβ. (3.15) To recover the QGT, we take the difference between these two equations and obtain (∂iu † p + iα u† p ) · (∂jup − iβ up) − (∂iu † p · up + iα)(u† p · ∂jup − iβ) = ∂iu † p · ∂jup + i(αu† p · ∂jup − β∂iu † p · up) + αβ − [ (∂iu † p · up)(u† p · ∂jup) + i(αu† p · ∂jup − β∂iu † p · up) + αβ ] = ∂iu † p · ∂jup − (∂iu † p · up)(u† p · ∂jup) (3.16) 34 Finally, we can set α or β to be, either t ∂mEp or ∂pzEp + z. Thus, we conclude that the QGT for the free Dirac equation is Qij [up] = ⟨∂iψ|ψjψ⟩ − ⟨∂iψ|ψ⟩⟨ψ|∂jψ⟩ = ⟨∂iup|∂jup⟩ − ⟨∂iup|up⟩⟨up|∂jup⟩. (3.17) This demonstrates that the quantum geometric tensor for free Dirac spinors is fully captured by the spinor structure, independent of the plane-wave factor. In the following sections, we apply this result to explicit computations in Dirac and Weyl representations 3.2 QGT Dirac representation (λ = {m}). The γ matrices for the Dirac representation are given by γ0 = ( I 0 0 −I ) & γi = ( 0 σi −σi 0 ) , (3.18) where σi are the Pauli matrices: σ1 = ( 0 1 1 0 ) , σ2 = ( 0 −i i 0 ) , σ3 = ( 1 0 0 −1 ) . (3.19) Then, the Dirac equation (3.1) in matrix form is     p0 −m 0 p3 p1 − ip2 0 p0 −m p1 + ip2 −p3 −p3 −p1 + ip2 −p0 −m 0 −p1 − ip2 p3 0 −p0 −m         u1(p) u2(p) v1(p) v2(p)     =     0 0 0 0     (3.20) The normalized solutions to this system of equations are: Particles (positive-energy-solutions): u1(p) = √ (m+ p0) 2p0     1 0 − p3 p0+m −p1+ip2 p0+m     & u2(p) = √ (m+ p0) 2p0     0 1 −p1−ip2 p0+m p3 p0+m     . (3.21) Anti-particles (negative-energy-solutions): v1(p) = √ p0 −m 2p0     p3 m−p0 p1+ ip2 m−p0 1 0     & v2(p) = √ p0 −m 2p0     p1−ip2 m−p0 − p3 m−p0 0 1     (3.22) Let’s consider the inner product2 ⟨ui|uj⟩ = u†iuj = δij = v†i vj = ⟨vi|vj⟩ (3.23) where i = 1, 2. Consider the particles solution ui to calculate the QGT taking as the only parameter to be the mass of the system, i.e. λ = m First, we compute the quantities: ⟨∂mui|∂mui⟩ = ∂mu † i ∂mui = m2 + 2mp0 + p20 + p⃗ 2 8p0(m+ p0)3 (3.24) 2u† represents the transpose conjugate of u. 35 ⟨∂mui|ui⟩ = −−m2 − 2mp0 − p20 + p⃗ 2 4p0(m+ p0)2 = ⟨ui|∂mui⟩ (3.25) where p⃗ 2 = p21 + p22 + p23. Then, the QGT for the particle ui(p) is Qmm[ui] = ⟨∂mui|∂mui⟩ − ⟨∂mui|ui⟩⟨ui|∂mui⟩ = 2mp0 ( 3p⃗ 2 −m2 ) − ( p⃗ 2 −m2 )2 + 2mp30 + p40 + 4p⃗ 2p20 16p20 (m+ p0) 4 . (3.26) We need to consider the relation dispersion p20 − p⃗ 2 = m2 (3.27) so that, the QGT for the particle ui(p) reduces to Qmm[ui] = p0 −m 4p20 (m+ p0) (3.28) For the antiparticles vi(p) we calculate the quantities: ⟨∂mvi|∂mvi⟩ = m2 − 2mp0 + p20 + p⃗ 2 8p0(p0 −m)3 (3.29) ⟨∂mvi|vi⟩ = −m2 + 2mp0 − p20 + p⃗ 2 4p0(m− p0)2 = ⟨vi|∂mvi⟩ (3.30) After considering the relation dispersion (3.27) the QGT for the antiparticle vi(p) becomes Qmm[vi] = m+ p0 4p20(p0 −m) (3.31) It is easy to see that making the transformation p0 → −p0 from particles to antiparticles the QGT are the same. 36 Quantum Geometric Tensor, λ = m QGT vs mass m. 2 4 6 8 10 12 m 0.0005 0.0010 0.0015 0.0020 0.0025 Figure 3.1: The QGT is a monotonically decreasing function, which becomes zero when the mass equals the energy (p0). When the mass is larger than the energy, it becomes negative, and the validity of the QGT breaks, since at this critical point, the disper- sion relation (3.42) does not hold anymore. QGT vs Energy p0. 2 4 6 8 10 p0 0.005 0.010 0.015 0.020 Figure 3.2: The QGT has a maximum. A deeper analysis using the fidelity approach indicates that for values of p0 < p0−max, the approximation deviates from the full fidelity. At higher energies, the QGT provides a better approximation, tending to 1. This makes sense since at higher energies, the system can transition to the other state. QGT of particles and anti-particles in Dirac representation. Figure 3.3: The QGT of the particles (blue) and antiparticles (red) varying the mass and the energy. We see that for antiparticles, p0 ≤ 0, and for particles, p0 ≥ 0. 37 3.3 QGT Weyl representation (λ = {m}). In Weyl representation the γ matrices are γ0 = ( 0 I I 0 ) γi = ( 0 −σi σi 0 ) (3.32) where σi are the Pauli matrices. Consider the Ansatz ψ(x, t) = u(p)e−ipµx µ for the solutions of the Dirac equation, then in matrix form we have     −m 0 p0 − p3 −p1 + ip2 0 −m −p1 − ip2 p0 + p3 p0 + p3 p1 − ip2 −m 0 p1 + ip2 p0 − p3 0 −m     u(p) = 0. (3.33) where the normalized solutions are given for particles: u1(p) = A     m+ p0 − p3 −p1 − ip2 p3 +m+ p0 p1 + ip2     u2(p) = A     −p1 + ip2 p3 + p0 +m p1 − ip2 m+ p0 − p3     (3.34) and for antiparticles v1(p) = B     m+ p0 − p3 −p1 − ip2 −(p3 +m+ p0) −(p1 + ip2)     v2(p) = B     −p1 + ip2 p3 + p0 +m −p1 + ip2 −(m+ p0 − p3)     (3.35) where the normalization constants are the same for particles and anti-particles: A = 1√ 2 √ (m+ p0)2 + p⃗ 2 = B. (3.36) We consider the inner product define by (3.23). Taking as the parameter λ to be only the mass m, then the QGT for the particles, ui(p), are: Qmm[ui] = p21 + p22 + p23 (m2 + 2mp0 + p20 + p21 + p22 + p23)2 = p0 −m 4p20(m+ p0) (3.37) which is the same as the one found for the Dirac representation! Now, for the antiparticles we have that the QGT is Qmm[vi] = p21 + p22 + p23 (m2 + 2mp0 + p20 + p21 + p22 + p23)2 = p0 −m 4p20(m+ p0) (3.38) which is the same as for the particles. This is in accordance that particles and antiparticles in Weyl representation both have p0 > 0, and to distinguish them, we need to check how they transform under charge conjugation. 38 3.4 Fidelity approach. A fundamental aspect of quantum computing is assessing the difference between an input state |in⟩ and an output state |out⟩. For mixed states, this comparison is carried out using the density matrix formalism. However, in the case of pure states, the density matrix reduces to a simpler expression involving the quantum fidelity, which we introduce here. We aim to compare this quantity with the QGT in spin- 12 systems. Recall the definition of fidelity (2.18): F (λ′, λ) = |⟨ψ(λ′)|ψ(λ)⟩|. (3.39) 3.4.1 Dirac Representation. For the case of particles ψ(λ) → ui(m), i = 1, 2. Let’s consider λ′ = m and λ = m0, then we have that the fidelity is given by3 ⟨ui(m)|ui(m0)⟩ = √ [(m+ p0)(m0 + p0) + p21 + p22 + p23] 2 (m2 + 2mp0 + p20 + p21 + p22 + p23)(m2 0 + 2m0p0 + p20 + p21 + p22 + p23) (3.40) which up to third order ⟨ui(m+ ∆m)|ui(m)⟩ =1 − 1 2 (p21 + p22 + p23) (m2 + 2mp0 + p20 + p21 + p22 + p23)2 (∆m)2 − mp21 + p0p 2 1 +mp22 + p0p 2 2 +mp23 + p0p 2 3 (m2 + 2mp0 + p20 + p21 + p22 + p23)3 (∆m)3 + O [ ∆m4 ] (3.41) with ∆m an infinitesimal increment of the mass m. Applying the dispersion relations m2 = p20 − p⃗ 2 & m2 0 = p20 − p⃗ 2 (3.42) then, the fidelity becomes ⟨ui(m)|ui(m0)⟩ = 1 2 √ √ √ √ [ (m+ p0)(m0 + p0) + √ m2 − p20 √ m2 0 − p20 ]2 p20(m+ p0)(m0 + p) (3.43) and the susceptibility χD = (p21 + p22 + p23) (m2 + 2mp0 + p20 + p21 + p22 + p23)2 → −m+ p0 4p2(m+ p0) (3.44) which, as expected, is in accordance to the one obtained via the QGT . For the case of antiparticles after applying the dispersion relations (3.42) we find that: ⟨vi(m)|vi(m0)⟩ = 1 2 √ √ √ √ − [ (m− p0)(m0 − p0) + √ p20 −m2 √ p20 −m2 0 ]2 p20(m− p0)(p0 −m0) . (3.45) It is easy to show that the susceptibility is the same as the QGT for antiparticles, just as for the particles. 3We abuse the notation for ui(m) and vi(m) for the particles and antiparticles, respectively, where the dependence on the parameter m is given, instead of the usual dependence of the momentum p. 39 3.4.2 Weyl Representation. We can obtain the fidelity for the Weyl representation in the same way as for the Dirac representation. We need to note that in Weyl representation the fidelity for particles and antiparticles is the same. Thus, we find that the fidelity is given by ⟨ui(m)|ui(m0)⟩ = √ [(m+ p0)(m0 + p0) + p21 + p22 + p23] 2 (m2 + 2mp0 + p20 + p21 + p22 + p23)(m2 0 + 2m0p0 + p20 + p21 + p22 + p23) (3.46) which up to third order ⟨ui(m+ ∆m)|ui(m)⟩ = 1 − 1 2 (p21 + p22 + p23) (m2 + 2mp0 + p20 + p21 + p22 + p23)2 (∆m)2 − mp21 + p0p 2 1 +mp22 + p0p 2 2 +mp23 + p0p 2 3 (m2 + 2mp0 + p20 + p21 + p22 + p23)3 (∆m)3 + O [ ∆m4 ] (3.47) with ∆m an infinitesimal increment of the mass m. Applying the dispersion relations (3.42), we have ⟨ui(m)|ui(m0)⟩ = 1 2 √ √ √ √ [ (m+ p0)(m0 + p0) + √ m2 − p20 √ m2 0 − p20 ]2 p20(m+ p0)(m0 + p) (3.48) and the susceptibility χW = (p21 + p22 + p23) (m2 + 2mp0 + p20 + p21 + p22 + p23)2 → −m+ p0 4p2(m+ p0) (3.49) Thus, confirming that both the quantum susceptibility and the QMT are the same. In the following plots, we compare the full calculation of the quantum fidelity, F (m,m0, p0) (in blue), with the approximation given by the quantum susceptibility (in orange): F (m,m0, p0) ≈ 1 − 1 2 Qmm(m, p0) · (m0 −m)2, where m0 and p0 are fixed parameters. Also we use the notation ∆m2 = (m0 −m)2. 40 Fidelity vs the QGT approximation for λ = m. Fixing energy (p0 = 1.5) and mass (m0 = 1.1). 1 - 1 2 QMT(m, p0 = 1.5) (1.1 -m)2 Fidelity(m, m0 = 1.1, p0 = 1.5) 0.2 0.4 0.6 0.8 1.0 1.2 1.4 m 0.92 0.94 0.96 0.98 1.00 Figure 3.4: We observe that the QGT provides a very good approximation when the varying mass is less than or equal to the fixed mass (m0), i.e., for m ≤ m0. At m = m0, both curves coincide at the value 1, but interestingly the QGT approximation remains at this limit, while the fidelity decreases below 1 for values of m larger than m0. Additionally, note that in this case the fixed mass is less than the system’s energy: m0 < p0. If m0 > p0, the fidelity becomes imaginary; this also holds true for values of the varying mass m greater than the energy. This behavior arises due to the dispersion relation (3.42). Fixing the masses m = 1 and m0 = 1.1. Fidelity(m = 1, m0 = 1.1, p) 1 - 1 2 QMT(m = 1, p)Δm2 5 10 15 20 p 0.99990 0.99992 0.99994 0.99996 0.99998 1.00000 Figure 3.5: In this case, we let the energy (p0) vary. The QGT also provides a good approxi- mation as the energy p0 increases. Moreover, as mentioned above, the maximum of the QGT (which corresponds to a minimum in this plot, since we are considering 1− 1 2 QMT∆m2) can be interpreted as a critical point where the approximation begins to be accurate.. To conclude this section, we point out that there is no Berry curvature, since we are considering only one parameter (λ = m). Because the Berry curvature is antisymmetric, it vanishes automatically in this case. 41 3.5 QGT for spin-1 2 particles with λ⃗ = (m, p3). In this case, we show only the results using the Dirac representation, because they are the same for both representations: Dirac and Weyl (chiral). Recall the solutions to the Dirac equation (3.20): ψ(x) = { ui(p) e −ipµxµ particles vi(p) e −ipµxµ antiparticles i = 1, 2. (3.50) Let us point out that, even though the solutions have an imaginary part, the Berry curvature is still zero. This makes sense, since the Hamiltonian for a free Dirac particle lacks the necessary parameter-dependent interactions to induce a non-trivial geometric phase, despite considering parameters like the mass (m) and the momentum (p3). Thus, the topology remains trivial for a free particle. To compute the QGT, we are not going to take into account the integration over x, since this will give us an infinity, which can be regularised by introducing a finite volume: ⟨ψi(x)|ψj(x)⟩ = ∫ d3xψ† i (x)ψj(x) ∼−→ ψ† i (x)ψj(x) (3.51) Then, the QGT, which is the same as the QMT, becomes a 2 × 2 symmetric matrix with entries:4 Qmm[ui] = p21 + p22 + p23 (m2 + 2mp0 + p20 + p21 + p22 + p23)2 = p0 −m 4p20(m+ p0) (3.52) Qmp3 [ui] = − (m+ p0)p3 (m2 + 2mp0 + p20 + p21 + p22 + p23)2 (3.53) Qp3p3 [ui] = m2 + 2mp0 + p20 + p21 + p22 (m2 + 2mp0 + p20 + p21 + p22 + p23)2 (3.54) There is no surprise that the component Qmm is the same as above, which was made clear after the use of the dispersion relation. The determinant of the QGT is detQ = p21 + p22 m2 + 2mp0 + p20 + p21 + p22 + p23 . (3.55) It is interesting to note that when both momenta p1 and p2 are zero, the determinant of the QGT is zero. This can be thought as when the dispersion relation p20 = m2 + p23 (3.56) holds. 4For the anti-particles, the QGT is the same, but we need to consider again that p0 → −p0 in this representation . 42 Plots of the Components of the QGT, λ⃗ = (m, pz) Plot of Qmm Figure 3.6: Component Qmm of the QGT which is the same as for the previous section. In this plot we take the parameters 0 ≤ m ≤ 10 and −10 ≤ pz ≤ 10. 3D plot of Qmpz Figure 3.7: This plot shows the bahaviour of the component Qmpz when both parameters vary. Plot of Qmpz 2 4 6 8 10 -0.003 -0.002 -0.001 0.001 0.002 0.003 Figure 3.8: Component Qmpz we vary the mass m for different values of the momentum: pz = −1 (blue), pz = 0 (yellow) and pz = 1 (green). Plot of Qmpz -10 -5 5 10 m -0.2 -0.1 0.1 0.2 pz Figure 3.9: Component Qmpz we vary pz for differ- ent values of the mass: m1 = 0.5 (blue), m2 = 1 (yellow) and m3 = 1.5 (green). 43 3D plot of Qpzpz Figure 3.10: This plot shows the bahaviour of the component Qmpz when both parameters vary. Plot of Qpzpz 2 4 6 8 10 m 0.02 0.04 0.06 0.08 0.10 0.12 Figure 3.11: Component Qpzpz we vary the mass m for different values of the momentum: pz = −1 (blue), pz = 0 (yellow) and pz = 1 (green). Plot of Qmpz -10 -5 5 10 pz 0.2 0.4 0.6 0.8 1.0 Figure 3.12: Component Qmpz we vary pz for dif- ferent values of the mass: m1 = 0.5 (blue), m2 = 1 (yellow) and m3 = 3 (green). Plots of the Determinant of the QGT, λ⃗ = (m, pz) 3D plot of detQGT Figure 3.13: 3D plot of the determinant of the QGT, when we let the parameters m and pz vary freely. Plot of detQ -4 -2 2 4 pz -0.00002 -0.00001 0.00001 0.00002 0.00003 Figure 3.14: The determinant of the QGT becomes negative when the values of the dispersion relation (3.42) does not hold anymore. The determinant of the QGT is zero exactly when the dispersion relation is equal. 44 Fidelity approach: λ⃗ = (m, p3). We are going to work only with solutions corresponding to particles: ψi(x) = ui(p)e −ipµxµ and with the inner product of (3.51). Then, we have F(m∗, p3∗;m, p3) = ⟨ui(p;m∗, p3∗)e−ip∗µx µ |ui(p;m, p3)e−ipµx µ⟩ = √ ((m∗ + p0)(m+ p0) + p21 + p22 + p3p3∗) 2 (m2 ∗ + 2m∗p0 + p20 + p⃗ 2 ∗)(m2 + 2mp0 + p20 + p⃗ 2) (3.57) where p∗ = (p0, p1, p2, p3∗). By taking the series expansion up to second order F(m∗, p3∗;m, p3) =1 − 1 2 m2 + 2mp0 + p20 + p21 + p22 (m2 + p23 + 2mp0 + p20 + p21 + p22 + p23)2 ∆p23 + p3(m+ p0) (m2 + 2mp0 + p20 + p21 + p22 + p23)2 ∆p3∆m − 1 2 p21 + p22 + p23 (m2 + 2mp0 + p20 + p21 + p22 + p23)2 ∆m2 1 − 1 2 Qp3p3∆p23 −Qmp3∆p3∆m− 1 2 Qmm∆m2 (3.58) where ∆p3 = (p3 − p3∗), ∆m = (m−m∗), and Qij are the components of the QGT. Plots of Fidelity, F(m′, p′z;m, pz) 3D plot of F(m′, p′z; 1, 1) Figure 3.15: Behaviour of the quantum fidelity when we leavem0 = pz constant. The highest value is given when m = m0, which is equal to one. For greater values of pz the fidelity diminishes. It seems to be an abrupt transition in the zone around negative values of pz and m = 0. Thus, there is a strong dependence of pz for the fidelity. 3D plot of F(m′, 1;m, 1) Figure 3.16: Behaviour of the quantum fidelity when we leave p′z = pz constant. The highest value is given when m = m0, which is equal to one. For greater values of m the fidelity diminishes. It seems to be an abrupt decay in the fidelity when at least one of the masses tends to zero. 45 3D plot of F(1, p′z, 1, pz) Figure 3.17: Behaviour of the quantum fidelity when the mass is kept constant, in this case, we took m = 1 = m0. It’s interesting to see that the fidelity depends on the direction (sign) of the momenta of the states. When the signs align the fidelity is greater, when the signs differ the fidelity falls to a minimum. 3D plot of F(m, 1; 1, pz) Figure 3.18: Behaviour of the quantum fidelity when we leave p′z,m0 constants. The highest value is given when m = m0, which is equal to one. For greater values of m and pz the fidelity diminishes, but it will tend to one again when pz has the same sign as p′z. The fidelity is lower when pz and p′z have different sign. Plot of F(m, pz∗;mi, pz∗) mpzFidPart(m, 1, 1, 1) mpzFidPart(m, 1, 1.5, 1) mpzFidPart(m, 1, 0.5, 1) 2 4 6 8 10 m 0.88 0.90 0.92 0.94 0.96 0.98 1.00 Figure 3.19: We take pz∗ = 1 a constant, and mi = 0.5(blue), 1(yellow), 1.5(green). It’s interesting to note that the least difference between m = 0 and mi the fidelity is greater and approaches 1 faster, but it decreases to a positive finite value when m → ∞, e.g. (0.8507, 0.9239, 0.9571) for the values of mi = (0.5, 1, 1.5), respectively. Thus, the limit is not the same, but the fidelity decreases less when the initial value of mi is bigger. 46 Plot of F(m′ ∗, p ′ z∗;m∗, pz) mpzFidPart(1, 1, 0.5, p) mpzFidPart(1, 1, 1, p) mpzFidPart(1, 1, 1.5, p) -10 -5 5 10 p 0.5 0.6 0.7 0.8 0.9 1.0 Figure 3.20: We take m′ ∗ = m∗ = 1 a constant, and p′z∗ = 0.5(blue), 1(yellow), 1.5(green). As mentioned in the previous plot in 3D, it’s inter- esting to note that the fidelity responds differently when the sign of pz is the same or different to p′z∗. In this case the limit for all values of p′z∗ is the same: (.9239, pz → ∞) and (0.3827, pz → ∞). This tells us that the fidelity in this case does not depend on m, while in the previous case it depended on the value of pz too. Plot of F(m′ ∗, p ′ z∗;m∗, pz) mpzFidPart(1, 0, 1, pz) mpzFidPart(1, 1, 1, pz) mpzFidPart(1, -1, 1, pz) -10 -5 5 10 pz 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Figure 3.21: This plot shows that, when m′ ∗ = m∗ = const, the fidelity is greatest (equals one) when pz = p′z∗. Also the dependence on the sign between pz and p′z∗ is relevant, when they differ the fidelity diminishes more, than when they are the same. Finally, when p′z∗ = 0 the fidelity is symmetric. * 3.6 QGT for a Weyl Semimetal. We will consider the most simple Weyl semimetal which its Hamiltonian is H = vf σ⃗ · k⃗ (3.59) 47 Plot of F(m′ ∗, pz;m∗, pz) mpzFidPart(1, pz, 0.5, pz) mpzFidPart(1, pz, 1, pz) mpzFidPart(1, pz, 1.5, pz) -10 -5 5 10 pz 0.985 0.990 0.995 1.000 Figure 3.22: In this plot we show the behaviour of the quantum fidelity when m′, and m are both constants and we vary the momentum pz. The fidelity is symmetric around the y-axis irrespective of the value of the masses. But the fidelity has a maximum when m′ = m, and minima where most likely some resonance between the states may occur. where vf is the Fermi velocity and k⃗ is the momenta in the reciprocal lattice. In matrix form is H = ( −k3vf −(k1 − ik2)vf −(k1 + ik2)vf k3vf ) . (3.60) Thus, we want to find the eigenvectors for this Hamiltonian: HΨ = EΨ. There are two eigenvectors given by ψ1 = A ( k3+ √ k2 1 +k2 2 +k2 3 k1+ik2 1 ) (3.61) and ψ2 = B ( k3− √ k2 1 +k2 2 +k2 3 k1+ik2 1 ) (3.62) where A = ( 1 2 − k3 2 √ k21 + k22 + k23 )1/2 (3.63) B = ( 1 2 + k3 2 √ k21 + k22 + k23 )1/2 . (3.64) Let’s consider the parameters to be k1, k2, and k3 is easy to find that the components of the QMT are Gii = 3 ∑ j=1 k2j − k2i 4k⃗ 4 (3.65) and Gij = −kikj 4k⃗ 4 (3.66) where k⃗ 2 = k21 + k22 + k23. 48 This system has a Berry curvature different from zero given by5 Fij = ϵijk kk 4k⃗ 3 (3.67) which is, and must be, the same as the one found in the literature of condensed matter. The QGT for the Weyl semimetal is singular, thus we set w.l.o.g. kx = const and work with the subdeterminant which is no longer singular: DetQMTyz = k2x 16(k2x + k2y + k2z)3 (3.68) In this case, we calculated the curvature of the parameter manifold, i.e. the Ricci scalar R, which is a positive constant: R = 8. (3.69) Recall, that the Ricci scalar R of an n-dimensional sphere of radius r, satisfies the relation R = n(n− 1) r2 (3.70) so, for our 2-dimensional case the radius of the sphere is r = 1√ 2 (3.71) Plots of the Subdeterminant of the QMT. Subdeterminant of the QMT, ∥k∥ = c Figure 3.23: The full QMT is singular. Thus, we consider w.l.o.g kx to be constant. Then, for differ- ent values of kx we consider the dispersion relation ∥k∥2 = k2x + k2y + k2z to be constant too. Subdeterminant of the QMT, kx = c Figure 3.24: kx is left as a constant and we let the parameters ky and kz take different values. The results obtained show that the parameter space defined by the vector k⃗ = (kx, ky, kz) for the Weyl semimetal has the geometry of a sphere of radius r = 1/ √ 2, that can be related to the Bloch sphere. 5Calculated using the QGT approach. 49 This geometrical structure arises from the quantum metric tensor, which captures the infinitesimal distances between quantum states in the parameter space. As expected, the Berry curvature computed from the QGT is non-zero and matches the topological structure associated with the Weyl nodes of the condensed matter theory. However, the Berry curvature leads to quantized topological invariants like the Chern number, while the QMT reveals the differential geometric structure, such as the curvature of the parameter space. This allows us to interpret the Bloch sphere as the effective manifold in which the Weyl fermions in the semimetal evolve. 50 Chapter 4 Quantum Geometric Tensor in Quantum Field Theories. Up until now, we have worked within the framework of quantum mechanics; that is, we have considered systems where the particles are treated as discrete entities. The next natural step is to consider fields, thus the question if it is possible to extend the Quantum Geometric Tensor (QGT) from quantum mechanics to quantum field theory (QFT) arises. The short answer is yes, it is possible. The path integral formalism allows us to extend the QGT to this more general case. One caveat of this formalism is that, so far, it has not been possible to define an object analogous to the Berry connection. This means that there is no known quantity whose exterior derivative yields the Berry curvature in the same way as in the quantum mechanical case [12]. On the positive side, however, the Berry curvature can still be computed via the commutator of certain operators, Oα. Also, the path integral formalism comes with several benefits: • By considering classical field configurations one obtains the quantum amplitude. • Classical fields commute, so there is no need to worry about commutation relations among fields. • Time ordering comes free. • Lorentz invariance is always manifest. • For perturbation theory, the Feynman diagrams emerge in a very natural way. Finally, the theory is governed by the (Euclidean) Lagrangian instead of the Hamiltonian, which might be considered a downgrade because the unitarity of the system is no longer immediately manifest. Considering these “benefits” and the fact that the canonical formalism has not yet been developed, we proceed to show the general construction of the QGT using the path integral formalism for d+1-dimensional theories, following [12,33]. 4.1 QGT in d+ 1 dimensional QFT’s: general theory. Let us consider a QFT initially described by the Euclidean Lagrangian L0, which depends explicitly on the parameters λα belonging to the parameter manifold M. Then, at time τ = 0, we turn on a deformation that perturbs the original Lagrangian by the deformation operators Oα, so that the final Lagrangian L1 is given by the original Lagrangian with the addition of Oαδλ α, which represent the infinitesimal changes of the parameters; that is: L1(ϕ, λ) = L0(ϕ, λ+ δλ) = L0(ϕ, λ) + ∂L0 ∂λα δλα +O ( (δλ)2 ) . (4.1) Therefore, we define the deformation operators associated with the variation of the parameters as Oα = ∂L ∂δλα . (4.2) 51 If we consider |Ω0⟩ and |Ω1⟩ to be the vacua (the states of minimal energy) associated with their respective theory described by L0 and L1, then the overlap between these two vacuum states is given by1 [12, 33] ⟨Ω1|Ω1⟩ = 1√ Z0Z1 ∫ Dϕ exp  − 0 ∫ −∞ dτ ∫ ddxL0 − ∞ ∫ 0 dτ ∫ ddxL1   (4.3) where Zi are the usual partition functions Zi = ∞ ∫ −∞ dτ ∫ ddx exp [−Li] . (4.4) Given that there are two vacua, it is noteworthy to point out that the expectation value of a given operator A is taken with respect to the original Lagrangian ⟨A⟩ = 1 Z0 ∫ DϕAe − ∞∫ −∞ dτ ∫ ddxL0 . (4.5) Inserting the perturbed Lagrangian L1 = L0 + Oαδλ α into equation (4.3), the overlap becomes ⟨Ω1|Ω2⟩ = 〈 exp ( − ∞ ∫ 0 dτ ∫ ddxOαδλ α )〉 〈 exp ( − ∞ ∫ −∞ dτ ∫ ddxOαδλα )〉1/2 (4.6) Assuming that the two-point functions have time-revearsal symmetry ⟨Oα(−τ1, x)Oβ(−τ2, x)⟩ = ⟨Oα(τ1, x)Oβ(τ2, x)⟩ (4.7) and that the discontinuity between going from the original Lagrangian to the perturbed one is of no concern2, we expand up to second order the fidelity between these two vacuum states: |⟨Ω1|Ω0⟩| = 1 − 1 2 Qαβδλ αδλβ + · · · (4.8) where Qαβ = 0 ∫ −∞ dτ1 ∫ ddx1 ∞ ∫ 0 dτ2 ∫ ddx2 [ ⟨Oα(τ1, x1)Oβ(τ2, x2)⟩ − ⟨Oα(τ1, x1)⟩⟨Oβ(τ2, x2)⟩ ] (4.9) is the Quantum Geometric Tensor in the path integral formalism.3 At this point, one might think that we are ready to tackle all QFT systems found in the literature, which may be true; however, we will inevitably encounter the same challenges they present. Furthermore, only a limited number of systems can be solved analytically. These concerns motivate us to turn our attention away from higher-dimensional or more complicated theories and instead focus on the simplest case: a theory with no spatial or temporal dimensions. 1Due to the deformation of L0, the original vacuum is no longer the vacuum state of the perturbed Lagrangian, thus the overlap between them is not trivial. This means that it is not equal to one anymore. 2To tackle this discontinuity, that can result in ultraviolet divergencies, is to introduce a cutoff scale around τ = 0. More in ?? and references therin. 3As in the standard case, the real part of this tensor is the QMT and the imaginary part corresponds to the Berry curvature. 52 4.2 QGT in 0 + 0 Dimensions. To build intuition about the QGT in higher dimensional quantum field theories, we begin by analyzing the simplest possible case: a theory in 0 + 0 dimensions. In this scenario, the field behaves as a real random variable in the interval (−∞,∞); that is, the field represents the probability density over a manifold that consists only of a single point that can take any value in R. We proceed to demonstrate that the transition amplitude between the vacua corresponding to the two different Lagrangians L0 and L1 mentioned above can be computed explicitly. The probability density associated with a quantum field ϕ, treated as a real random variable, is given by [34]: P(ϕ) = 1 Z0 e−S(ϕ). (4.10) The partition function is defined as Z0 = ∫ dϕe−S(ϕ) (4.11) where S(ϕ) is the action of the theory. Now is a good moment to state that in 0 + 0-theories the action S and the Lagrangian L are equivalent. Formally4, we can describe the probability density as |⟨q|ϕ⟩|2 = P(ϕ), (4.12) where |ϕ⟩ can be thought as a formal state of the continous basis of R labeled by ϕ, and |q⟩ the analog of the physical state defined by the exponential weight e− 1 2 S . Therefore, we express the probability density in the quantum state representation as ⟨q|ϕ⟩ = e− 1 2 S(ϕ) Z1/2 0 . (4.13) Now, we consider the two vacua |Ωi⟩ of the two different Lagrangians Li(ϕ) (or equivalently, actions Si(ϕ)), with i = 0, 1, and the completeness relation ∫ dϕ |ϕ⟩⟨ϕ| = 1, (4.14) to compute the transition amplitude between them: ⟨Ω1|Ω0⟩ = ∫ dϕ⟨Ω1|ϕ⟩⟨ϕ|Ω0⟩ = ∫ dϕ e− 1 2 S1(ϕ)e− 1 2 S0(ϕ) √ Z1Z0 . (4.15) Let L1(ϕ) be a perturbation of L0: L1(ϕ) = L0(ϕ) + Oαδλ α (4.16) and let the expectation value of an operator be given with respect to the original action (Lagrangian), which in this case means ⟨A⟩ = 1 Z0 ∫ dϕA e−S0(ϕ). (4.17) 4Recall that there is no true Hilbert space in 0 + 0, thus this representation is only useful to understand the transition amplitude between states. 53 Then, the overlap between the vacuum states is given by ⟨Ω1|Ω0⟩ = ∫ dϕ e− 1 2 L1(ϕ) √ Z1 e− 1 2 L0(ϕ) √ Z0 = 1√ Z1Z0 ∫ dϕe−L0e− 1 2 Oαδλ α = Z0√ Z1Z0 1 Z0 ∫ dϕe−L0e− 1 2 Oαδλ α = √ Z0 Z1 ⟨e− 1 2 Oαδλ α⟩ = ⟨e− 1 2 Oαδλ α⟩ ⟨e−Oαδλα⟩1/2 . (4.18) Therefore, the transition amplitude is defined in the same manner as for higher dimensional theories, but without the integrals over time and space: ⟨Ω1|Ω0⟩ = ⟨e− 1 2 Oαδλ α⟩ ⟨e−Oαδλα⟩1/2 (4.19) which makes the QGT to be well-defined in the same sense as equation (4.9). Indeed, let’s consider up to second order the expansion of (4.19). Numerator series expansion: ⟨e− 1 2 Oαδλ α⟩ = 1 − 1 2 ⟨Oα⟩δλα + 1 8 ⟨OαOβ⟩δλαδλβ +O(δλ3) (4.20) Denominator series expansion:5 ⟨e−Oαδλ α⟩−1/2 = ( 1 − ⟨Oα⟩δλα + 1 2 ⟨OαOβ⟩δλαδλβ )−1/2 = 1 − 1 2 ( ⟨Oα⟩δλα − 1 2 ⟨OαOβ⟩δλαδλβ ) + 3 8 ( ⟨Oα⟩δλα + 1 2 ⟨OαOβ⟩δλαδλβ )2 + · · · (4.21) Then, combining both expressions is easy to verify that ⟨Ω1|Ω0⟩ = 1 + 1 8 ( ⟨OαOβ⟩ − ⟨Oα⟩⟨Oβ⟩ ) δλαδλβ +O(δλ3). (4.22) Thus, the Quantum Geometric Tensor in 0 + 0 dimensions is Qαβ = 1 4 [ ⟨OαOβ⟩ − ⟨Oα⟩⟨Oβ⟩ ] (4.23) so that ⟨Ω1|Ω0⟩ = 1 − 1 2 Qαβδλ αδλβ (4.24) as it is usually defined. Aside: 5(1− x)−1/2 = 1 + 1 2 x+ 3 8 x2 +O(x3). 54 For the last equality in (4.18) we use √ Z0 Z1 = ( √ Z1 Z0 )−1/2 = ( 1 Z0 ∫ dϕe−Oαδλ α eL0(ϕ) )−1/2 = ( ⟨e−Oαδλ α⟩ )−1/2 . (4.25) 55 4.3 Quantum Geometric Tensor for the free theory with a source. In this section, we compute the QGT for the next simplest quantum field theory: the scalar field theory with the addition of a constant, nonzero source J . The Lagrangian for any dimension is6 L = 1 2 ∂µϕ∂µϕ+ α 2 ϕ2 + ϕJ (4.26) We define the “zero” partition function as: Z0 = Z[J = const]. (4.27) This choice introduces an important subtlety in the computation of the QGT, as we now have two parameters to consider: the mass term, α and the nonzero source J . We will consider the QGT to be given without the overall factor 1 4 . That is, we will consider: Qαβ = ⟨OαOβ⟩ − ⟨Oα⟩⟨Oβ⟩, (4.28) since the behaviour of the QGT is not affected by this factor. Nevertheless, we need to be cautions when comparing the components or the determinant of the QGT with those of higher dimensions; however, for the Ricci scalar, this factor does not contribute either. Before moving on with the explicit computation of the QGT components for higher dimensions, we present Wick’s theorem to simplify the calculations corresponding to the product of the deformation operators, since they are functions of the field ϕ. Furthermore, we have to emphasize that the expectation value of a field in this theory is different from zero; that is ⟨ϕ(x)⟩ ≠ 0. (4.29) Wick’s Theorem [35]: The time-order product of the product of n fields is given by the normal-order of the product of the fields plus all possible contractions of them: T{ϕ1 · · ·ϕn} = : ϕ1 · · ·ϕn + all possible contractions : (4.30) where : ϕiϕj : denotes the normal order of the two fields and we use the notation ϕ(x1) = ϕ1. This theorem together the generating functional allow us to write the expectation value of the products of the fields in terms of two-point Green functions or two connected fields. To this end, consider the following, which will be used in the remains calculations: ⟨Tϕiϕj⟩ = ⟨ϕiϕj⟩ + ⟨ϕi⟩⟨ϕj⟩, (4.31) ⟨Tϕiϕjϕk⟩ = ⟨ϕiϕj⟩⟨ϕk⟩ + ⟨ϕjϕk⟩⟨ϕi⟩ + ⟨ϕjϕk⟩⟨ϕi⟩ + ⟨ϕi⟩⟨ϕj⟩⟨ϕj⟩, (4.32) ⟨Tϕiϕjϕkϕℓ⟩ = ⟨ϕiϕj⟩⟨ϕk⟩⟨ϕℓ⟩ + ⟨ϕjϕk⟩⟨ϕℓ⟩⟨ϕi⟩ + ⟨ϕkϕℓ⟩⟨ϕi⟩⟨ϕj⟩ + ⟨ϕℓϕi⟩⟨ϕj⟩⟨ϕk⟩ (4.33) + ⟨ϕiϕk⟩⟨ϕj⟩⟨ϕℓ⟩ + ⟨ϕℓϕj⟩⟨ϕk⟩⟨ϕ1⟩ + ⟨ϕiϕj⟩⟨ϕkϕℓ⟩ + ⟨ϕiϕk⟩⟨ϕjϕℓ⟩ (4.34) + ⟨ϕℓϕj⟩⟨ϕiϕk⟩ (4.35) where the two-point Green (connected) function is given by7 ⟨ϕiϕj⟩ = ∫ dk0 2π ∫ ddk (2π)d eik0(τj−τi)+ik⃗·(x⃗j−x⃗i) k20 + k⃗2 + α . (4.36) Now we have all the tools to present explicit calculations and results for the system in 0 + 0, 1 + 0 and 1 + 1 dimensions. We only show the results for 1 + 2 and 1 + 3, since they follow straightforwardly from the steps of the previous 1 + d calculations. Interestingly, these results can be generalized to any temporal and spatial dimension; that is, for any t+ d-dimensional theory. Using these generalized expressions for the components of the QGT, we compute the determinant and the Ricci scalar. 6Recall that we are working with the Euclidean Lagrangian from the beginning. 7This is the standard Green function of the theory for 1 + d dimensions [35]. 56 4.3.1 QGT in 0 + 0 dimensions. In this case the Lagrangian is given by8 L = α 2 ϕ2 + ϕJ. (4.37) We compute the deformation operators with respect to the parameters α and J : Oα = ϕ2 2 & OJ = ϕ. (4.38) The partition function for this theory is Z0 = ∞ ∫ −∞ dϕ e− α 2 ϕ2−ϕJ . (4.39) Then, we show how to compute each of the components of the QGT. • Qαα: By the definition of the QGT and the deformation operators (4.38) Qαα = ⟨O2 α⟩ − ⟨Oα⟩2, (4.40) which gives Qαα = 1 4Z0 ∞ ∫ −∞ dϕϕ4e− α 2 ϕ2−ϕJ − 1 4Z2 0   ∞ ∫ −∞ dϕϕ2e− α 2 ϕ2−ϕJ   2 . (4.41) These integrals can be computed analytically, yielding: Qαα = 1 2α2 + J2 α3 . (4.42) • QαJ : Similarly, QαJ = ⟨OαOJ⟩ − ⟨Oα⟩⟨OJ⟩ = 1 2 ⟨ϕ3⟩ − 1 2 ⟨ϕ2⟩⟨ϕ⟩, (4.43) which can be written as QαJ = 1 2Z0 ∞ ∫ −∞ dϕϕ3e− α 2 ϕ2−ϕJ − 1 2Z2 0   ∞ ∫ −∞ dϕϕ2 e− α 2 ϕ2−ϕJ     ∞ ∫ −∞ dϕϕ e− α 2 ϕ2−ϕJ   . (4.44) The result is: QαJ = − J α2 . (4.45) • QJJ : Finally, QJJ = ⟨O2 J⟩ − ⟨OJ⟩2 = ⟨ϕ2⟩ − ⟨ϕ⟩2, (4.46) which reads QJJ = 1 Z0 ∞ ∫ −∞ dϕϕ2e− α 2 ϕ2−ϕJ − 1 Z2 0   ∞ ∫ −∞ dϕϕ e− α 2 ϕ2−ϕJ   2 . (4.47) The analytical result is: QJJ = 1 α . (4.48) 8The results for 0 + 0 up to 1 + 3 dimensions can be found at the beginning of the section: Generalization of the QGT components for any spatial and time dimensions. 57 4.3.2 QGT in 1 + 0 (Quantum Mechanics). The Lagrangian for the standard theory in quantum mechanics of a free particle with a source is L = 1 2 ϕ̇2 + α 2 ϕ2 + ϕJ, (4.49) so that, the action becomes S[ϕ] = ∞ ∫ −∞ dτ ϕ̇2 2 + α 2 ϕ2 + ϕJ (4.50) In this case, the QGT (4.9) is given by Qαβ = 0 ∫ −∞ dτ1 ∞ ∫ 0 dτ2 [⟨Oα(τ1)Oβ(τ2)⟩ − ⟨Oα(τ1)⟩⟨Oβ(τ2)⟩] (4.51) Then the components of the QGT can be computed as follows: • Qαα: Qαα = 0 ∫ −∞ dτ1 ∞ ∫ 0 dτ2 1 4 [ ⟨ϕ2(τ1)ϕ2(τ2)⟩ − ⟨ϕ2(τ1)⟩⟨ϕ2(τ2)⟩ ] (4.52) To simplify the calculations we take ϕi = ϕ(τ1) and by following Wick’s theorem with the proper substi- tutions in (4.31) we have the following equations ⟨ϕ21⟩⟨ϕ22⟩ = ⟨ϕ1ϕ1⟩⟨ϕ2ϕ2⟩ + ⟨ϕ1ϕ1⟩⟨ϕ2⟩2 + ⟨ϕ2ϕ2⟩⟨ϕ1⟩2 + ⟨ϕ1⟩2⟨ϕ2⟩2 (4.53) and ⟨ϕ21ϕ22⟩ =⟨ϕ1ϕ1⟩⟨ϕ2ϕ2⟩ + 2⟨ϕ1ϕ2⟩2 + ⟨ϕ1ϕ1⟩⟨ϕ2⟩2 + 4⟨ϕ1ϕ2⟩⟨ϕ1⟩⟨ϕ2⟩ + ⟨ϕ2ϕ2⟩⟨ϕ1⟩2 + ⟨ϕ1⟩2⟨ϕ2⟩2. (4.54) Then, the term inside the brakets of equation (4.52) becomes ⟨ϕ21ϕ22⟩ − ⟨ϕ21⟩⟨ϕ22⟩ = 2⟨ϕ1ϕ2⟩⟨ϕ1ϕ2⟩ + 4⟨ϕ1ϕ2⟩⟨ϕ1⟩⟨ϕ2⟩. (4.55) In this manner, it follows that Qαα = 0 ∫ −∞ dτ1 ∞ ∫ 0 dτ2 [ 1 2 ⟨ϕ1ϕ2⟩⟨ϕ1ϕ2⟩ + ⟨ϕ1ϕ2⟩⟨ϕ1⟩⟨ϕ2⟩ ] (4.56) 58 To compute the integrals of the expression above, let’s take the first term: I1 : = 1 2 0 ∫ −∞ dτ1 ∞ ∫ 0 dτ2 1 2 ⟨ϕ1ϕ2⟩⟨ϕ1ϕ2⟩ = 1 2 0 ∫ −∞ dτ1 ∞ ∫ 0 dτ2 ∞ ∫ −∞ dk0 2π eik0(τ2−τ1) k20 + α ∞ ∫ −∞ dk′0 2π eik ′ 0(τ2−τ1) k ′2 0 + α = 1 2 ∞ ∫ −∞ dk0dk ′ 0 (2π)2 0 ∫ −∞ dτ1 ∞ ∫ 0 dτ2 ei(k0+k ′ 0)(τ2−τ1) (k20 + α)(k ′2 0 + α) = 1 2 ∞ ∫ −∞ dk0dk ′ 0 (2π)2 −1 (k20 + α)(k ′2 0 + α)(k0 + k′0)2 = 1 32α2 . (4.57) Now, working with the second term I2 : = 0 ∫ −∞ dτ1 ∞ ∫ 0 dτ2⟨ϕ1ϕ2⟩⟨ϕ1⟩⟨ϕ2⟩ = J2 0 ∫ −∞ dτ1 ∞ ∫ 0 dτ2 ∞ ∫ −∞ dτ ′dτ ′′ ∞ ∫ −∞ dk0dk ′ 0dk ′′ 0 (2π)3 eik0(τ2−τ1)eik ′ 0(τ ′−τ1)eik ′′ 0 (τ ′′−τ2) (k20 + α)(k ′2 0 + α)(k ′′2 0 + α) = J2 0 ∫ −∞ dτ1 ∞ ∫ 0 dτ2 ∞ ∫ −∞ dk0dk ′ 0dk ′′ 0 (2π)3 eik0(τ2−τ1)δ(k′0)δ(k′′0 ) (k20 + α)(k ′2 0 + α)(k ′′2 0 + α) = J2 ∞ ∫ −∞ dk0 2π 0 ∫ −∞ dτ1 ∞ ∫ 0 dτ2 eik0(τ2−τ1) α2(k20 + α) = −J2 ∞ ∫ −∞ dk0 2π 1 α2k20(k20 + α) = J2 2α7/2 (4.58) Therefore the Qαα component of the QGT is: Qαα = 1 32α2 + J2 2α7/2 (4.59) Note: To compute the last integral of I2, we need to compute first the indefinite integral and, then, take the corre- sponding limits to get the result;that is −J2 ∫ dk0 2π 1 α2k20(k20 + α) = J2(α1/2 + k0 arctan ( k0 α1/2 ) ) 2πk0α1/2 Now, we continue with the calculation of the cross-term of the QGT QαJ . 59 • QαJ : QαJ = 0 ∫ −∞ dτ1 ∞ ∫ 0 dτ2 [ ⟨ϕ(τ1)2ϕ(τ2)⟩ − ⟨ϕ2(τ1)⟩⟨ϕ(τ2)⟩ ] (4.60) Using previous notation we have that ⟨ϕ21ϕ2⟩ = ⟨ϕ1ϕ1⟩⟨ϕ2⟩ + 2⟨ϕ1ϕ2⟩⟨ϕ1⟩ + ⟨ϕ1⟩2⟨ϕ2⟩ (4.61) and ⟨ϕ21⟩⟨ϕ2⟩ = ⟨ϕ1ϕ1⟩⟨ϕ2⟩ + ⟨ϕ1⟩2⟨ϕ2⟩. (4.62) Therefore, QαJ = 0 ∫ −∞ dτ1 ∞ ∫ 0 dτ2 ⟨ϕ1ϕ2⟩⟨ϕ1.⟩ = −J 0 ∫ −∞ dτ1 ∞ ∫ 0 dτ2 ∞ ∫ −∞ dτ ′ ∞ ∫ −∞ dk0dk ′ 0 (2π)2 eik0(τ2−τ1) k20 + α eik ′ 0(τ ′−τ1) k′20 + α = −J 0 ∫ −∞ dτ1 ∞ ∫ 0 dτ2 ∞ ∫ −∞ dk0dk ′ 0 (2π)2 eik0(τ2−τ1) k20 + α e−ik ′ 0τ1δ(k′0) (k ′2 0 + α) = ∞ ∫ −∞ dk0 2π 0 ∫ −∞ dτ1 ∞ ∫ 0 dτ2 eik0(τ2−τ1) α(k20 + α) = J ∞ ∫ −∞ dk0 2π 1 αk0(k20 + α) = − J 2α5/2 (4.63) Finally, the last term is • QJJ : QJJ = 0 ∫ −∞ dτ1 ∞ ∫ 0 dτ2 [⟨ϕ(τ1)ϕ(τ2)⟩ − ⟨ϕ(τ1)⟩⟨ϕ(τ2)⟩] (4.64) Since, ⟨ϕ1ϕ2⟩ = ⟨ϕ1ϕ2⟩ + ⟨ϕ1⟩⟨ϕ2⟩ (4.65) we have that QJJ = 0 ∫ −∞ dτ1 ∞ ∫ 0 dτ2 ⟨ϕ1ϕ2⟩ = 0 ∫ −∞ dτ1 ∞ ∫ 0 dτ2 ∞ ∫ −∞ dk0 2π eik0(τ2−τ1) k20 + α = − ∞ ∫ −∞ dk0 2π 1 (k20 + α)k20 = 1 2α3/2 (4.66) 60 4.3.3 QGT in 1 + 1 dimensions. Since we are in 1 + 1 dimensions we have that ϕi = ϕ(τi, xi). (4.67) The Euclidean Lagrangian is given by L = −1 2 ∂µϕ∂µϕ− α 2 ϕ2 − Jϕ (4.68) where we wrote the indices below just to emphasize the Euclidean metric and the action is S[ϕ] = ∫ dτdx [ −1 2 ∂µϕ∂µϕ− α 2 ϕ2 − Jϕ ] . (4.69) So, in this case the QGT is given by Qαβ = 0 ∫ −∞ dτ1 ∞ ∫ −∞ dx1 ∞ ∫ 0 dτ2 ∞ ∫ −∞ dx2 [⟨Oα(τ1, x1)Oβ(τ2, x2)⟩ − ⟨Oα(τ1, x1)⟩⟨Oβ(τ2, x2)⟩] (4.70) • Qαα Qαα = 0 ∫ −∞ dτ1 ∞ ∫ −∞ dx1 ∞ ∫ 0 dτ2 ∞ ∫ −∞ dx2 1 4 [ ⟨ϕ2(τ1, x1)ϕ2(τ2, x2)⟩ − ⟨ϕ2(τ1, x1)⟩⟨ϕ2(τ2, x2)⟩ ] (4.71) Applying the same notation as above and Wick’s Theorem, we arrive at a similar equation as (4.56) Qαα = 0 ∫ −∞ dτ1 ∞ ∫ −∞ dx1 ∞ ∫ 0 dτ2 ∞ ∫ −∞ dx2 [ 1 2 ⟨ϕ1ϕ2⟩⟨ϕ1ϕ2⟩ + ⟨ϕ1ϕ2⟩⟨ϕ1⟩⟨ϕ2⟩ ] . (4.72) Following the previous steps, we define and compute the following integrals: I1 : = 1 2 0 ∫ −∞ dτ1 ∞ ∫ −∞ dx1 ∞ ∫ 0 dτ2 ∞ ∫ −∞ dx2 ⟨ϕ1ϕ2⟩⟨ϕ1ϕ2⟩ = 1 2 0 ∫ −∞ dτ1 ∞ ∫ −∞ dx1 ∞ ∫ 0 dτ2 ∞ ∫ −∞ dx2 ∫ V ol dk0dkx (2π)2 eik0(τ2−τ1)+ikx(x2−x1) k20 + k2x + α ∫ V ol dk′0dk ′ x (2π)2 eik ′ 0(τ2−τ1)+ik ′ x(x2−x1) (k ′2 0 + k′2 x + α) = 1 2(2π)2 0 ∫ −∞ dτ1 ∞ ∫ 0 dτ2 ∫ V ol dk0dkx ∫ V ol dk′0dk ′ x ei(k0+k ′ 0)(τ2−τ1) (k20 + k2x + α) δ2(kx + k′x) (k ′2 0 + k′2 x + α) = 1 2(2π)2 ∫ V ol dk0dkx ∞ ∫ −∞ dk′0 0 ∫ −∞ dτ1 ∞ ∫ 0 dτ2 ei(k0+k ′ 0)(τ2−τ1) (k20 + k2x + α) δ(0) (k ′2 0 + k2x + α) = 1 2(2π)2 ∫ V ol dk0dkx ∞ ∫ −∞ dk′0 −δ(0) (k20 + k2x + α)(k ′2 0 + k2x + α)(k0 + k′0)2 = ∞ ∫ −∞ dkx δ(0) 32(k2x + α)2 = πδ(0) 64α3/2 (4.73) 61 If we consider a finite one-dimensional volume we have δ(0) = L 2π (4.74) such that, I1 = L 128α3/2 . (4.75) Now, we compute the second term given by I2 : = 0 ∫ −∞ dτ1 ∞ ∫ −∞ dx1 ∞ ∫ 0 dτ2 ∞ ∫ −∞ dx2 ⟨ϕ1ϕ2⟩⟨ϕ1⟩⟨ϕ2⟩ = 0 ∫ −∞ dτ1 ∞ ∫ −∞ dx1 ∞ ∫ 0 dτ2 ∞ ∫ −∞ dx2 ∫ V ol dk0dkx (2π)2 eik0(τ2−τ1)+ikx(x2−x1) (k20 + k2x + α) × J ∫ V ol dτ ′dx′ dk′0dk ′ x (2π)2 eik ′ 0(τ ′−τ1)+ik′x(x ′−x1) (k ′2 0 + k′2 x + α) J ∫ V ol dτ ′′dx′′ dk′′0dk ′′ x (2π)2 eik ′′ 0 (τ ′′−τ2)+ik′′x (x′′−x2) (k ′′2 0 + k′′2 x + α) = J2 0 ∫ −∞ dτ1 ∞ ∫ −∞ dk0 ∞ ∫ −∞ dkx ∞ ∫ 0 dτ2 ∞ ∫ −∞ dk′x eik0(τ2−τ1)δ(kx)2 α2(k20 + k2x + α) = −J2 ∞ ∫ −∞ dk0 α2 δ(0) k20(k20 + α) = J2L 2α7/2 . (4.76) where we used similar techniques shown in the previous section, specially in equation (4.74). Thus, Qαα = L 128α3/2 + J2L 2α7/2 . (4.77) • QαJ QαJ = 0 ∫ −∞ dτ1 ∞ ∫ −∞ dx1 ∞ ∫ 0 dτ2 ∞ ∫ −∞ dx2 [ ⟨ϕ(τ1, x1)2ϕ(τ2, x2)⟩ − ⟨ϕ2(τ1, x1)⟩⟨ϕ(τ2, x2)⟩ ] (4.78) which reduces to 62 QαJ = 0 ∫ −∞ dτ1 ∞ ∫ −∞ dx1 ∞ ∫ 0 dτ2 ∞ ∫ −∞ dx2 ⟨ϕ1ϕ2⟩⟨ϕ1⟩ = −J 0 ∫ −∞ dτ1 ∞ ∫ −∞ dx1 ∞ ∫ 0 dτ2 ∞ ∫ −∞ dx2 ∫ V ol dk0dkx (2π)2 eik0(τ2−τ1)+ikx(x2−x1) (k20 + k2x + α) × ∫ V ol dτ ′dx′ k′0dk ′ x (2π)2 eik ′ 0(τ ′−τ1)+ik′x(x ′−x1) (k ′2 0 + k′2 x + α) = −J 0 ∫ −∞ dτ1 ∞ ∫ 0 dτ2 ∞ ∫ −∞ dk0 ∞ ∫ −∞ dkx eik0(τ2−τ1)δ(kx)2 α(k20 + k2x + α) = J ∞ ∫ −∞ dk0 δ(0) α(k20 + α) = − JL 2α5/2 . (4.79) where we have set a finite volume L to avoid the divergence of the δ(0)-function. Finally, the last component • QJJ : QJJ = 0 ∫ −∞ dτ1 ∞ ∫ −∞ dx1 ∞ ∫ 0 dτ2 ∞ ∫ −∞ dx2 ⟨ϕ1ϕ2⟩ = 0 ∫ −∞ dτ1 ∞ ∫ −∞ dx1 ∞ ∫ 0 dτ2 ∞ ∫ −∞ dx2 ∫ V ol dk0dkx (2π)2 eik0(τ2−τ1)+ikx(x2−x1) (k20 + k2x + α) = 0 ∫ −∞ dτ1 ∞ ∫ 0 dτ2 ∞ ∫ −∞ dk0 ∞ ∫ −∞ dkx eik0(τ2−τ1)δ 2(kx) (k20 + k2x + α) = − ∞ ∫ −∞ δ(0) k20(k20 + α) = L 2α3/2 (4.80) 4.3.4 Generalization of the QGT components for any spatial and time dimen- sions. Let’s write the previously obtained results for the components of the QGT of the scalar free theory with a source in various dimensions in a compact form: where L is the size of the box to avoid the divergence of the δ-function. It is possible to generalized these results for any spatial dimension, d, and any time dimension t in the following manner: 63 Qαα =                        1 2α2 + J2 α3 0+0 1 32α2 + J2 α7/2 1+0 L 128α3/2 + J2L 2α7/2 1+1 L2 256πα + J2L2 2α7/2 1+2 L3 512π3/2α1/2 + J2L3 2α7/2 1+3 QαJ =                        − J α2 0+0 − J 2α5/2 1+0 − JL 2α5/2 1+1 − JL2 2α5/2 1+2 − JL3 2α5/2 1+3 QJJ =                        1 α 0+0 1 2α3/2 1+0 L 2α3/2 1+1 L2 2α3/2 1+2 L3 2α3/2 1+3 Generalization of the QGT components for any time and spatial dimensions: Qαα = Γ[ 4−d2 ]Ld 21+4t+dπd/2α 4−d 2 + J2Ld 2tα 6+t 2 , (4.81) QαJ = − JLd 2tα 4+t 2 , (4.82) QJJ = Ld 2tα 2+t 2 . (4.83) For our cases of interest (t = 0, 1 and d = 0, 1, 2, 3), the components diverge when α, the mass term, goes to zero, which indicates a phase transition where the theory becomes masless and instead of being quadratic becomes linear too. We have to mention the dependence on L for the three components 9 is the same. This, also depends on the spatial dimension of the theory, such that, for d = 0 there is no dependence on it. In the following plots we use L ̸= 0 to appreciate this dependence. For Qαα it is important to note that it depends quadratically on J , for any time or spatial dimensions. However, the scaling of the matter term α does depend on the spatial dimension d and on the time dimension t. For the spatial dimension we observe that for d > 4 instead of beign inversely proportional becomes proportional to it, i.e, for d < 4 Qαα ∼ 1 αr , meanwhile for d > 0 Qαα ∼ αr ′ , where r, r′ > 0. Now, for the time dimension t, the depence is always inversely proportional on α. The case of the spatial dimension, d, is also very interesting since it is inside the Γ function, which tells us that for d = 2k, for k > 2 and k ∈ Z + the Qαα component diverges, that is, for every even integer d ≥ 4, this component diverges, which might tell us that in this dimensions the theory is not valid or one needs to apply a dimensional regularization method. For the components QαJ and QJJ the dependence of the time dimension is given in the mass term, α which, in turn, is inversely proportional to it. The dependence of J is linear for QαJ (negative straight lines which pass through the origin), however, QJJ does not depend on J whatsoever. 9In this case there are only three independent components since QαJ = QJα. 64 Plots of the Qαα[α, J, L, t, d] component.10 3D plot of Qαα Figure 4.1: Here, we present 3D plots of Qαα while keeping the time and spatial dimensions fixed. An interesting observation is that the plot for the 1 + 0 case lies below the 0 + 0 case. This result may be related to the scalar curvature, where we find that R0+0 < R1+0. We also fix the spatial volume by setting the system size to L = 5, which highlights the dependence of Qαα on the spatial dimension d. Plot of Qαα, fixed J . Figure 4.2: This plot shows the projection of Qαα with J = 1. Additionally, we set L = 2 to enhance the visual distinction between the plots. Plot of Qαα, fixed α. Figure 4.3: This plot, with α fixed, shows the quadratic dependence on J , forming parabolas. As seen above, the 1 + 0 case lies below the 0 + 0 case. 65 Plots of the QαJ [α, J, L, t, d] component. 3D plot of QαJ Figure 4.4: 3D plots of the QαJ component for dif- ferent time and spatial dimensions of interest, allow- ing the parameters α and J to vary. We set L = 3 to enhance the distinguishability of the plots. Plot of QαJ , fixed J . Figure 4.5: Plot of the projection of QαJ , where we let the mass parameter α vary. When J < 0, the plots lie above the α-axis, i.e., they are positive, while for J > 0, they lie below. Plot of QαJ , fixed α. Figure 4.6: Projection of QαJ with α fixed. These plots are straight lines with a slope proportional to −J . 66 Plot of the QJJ [α, L, t, d] component. QJJ , varying α. Figure 4.7: Due to the fact that QJJ does not depend on J explicitly, we show the two-dimensional plots of its dependence on the mass related parameter α. 4.3.5 Determinant of the QGT for a scalar free theory with a source. With the general formulae of the components of the QGT, it is possible to calculate its determinant for any time and spatial dimension: det(Q) = Γ [ 4−d 2 ] L2d 21+5t+dπd/2α 6t−d 2 . (4.84) There are two interesting results which follow from the components of the QGT. The first one is that the factor Γ [ 4−d 2 ] is still there, thus for the values of d = 2k, where k is an even positive integer greater than two, the determinant diverges. The second one, is that the determinant does not depend on the parameter J . Instead, the determinant shows a dependence on the volume L of the system. The mass parameter α has an interesting behaviour with respect to the spatial and time dimensions, which, for our cases of interest is ∼ 1 αr with r a positive integer, but it would be possible that this factor becomes αr ′ with r′ positive for d < 6t. 67 Plots of the determinant of the QGT. 3D plot of det(Q)[α,L, t, d] Figure 4.8: This 3D plot of the determinant assumes L = 1, and we consider the time and spatial dimen- sions to be continuous. It is evident that for values where Γ [ 4−d 2 ] is indeterminate, the determinant is also indeterminate. Plot of det(Q), fixed L. Figure 4.9: This plot shows the α-dependence of the determinant of the QGT. One can observe that as α → 0, the determinant diverges, indicating a quan- tum phase transition related to the transition from a massive system to a massless one. Plot of det(Q), fixed α. Figure 4.10: This plot shows the dependence of the volume Ld in the theory. When we fix the spatial dimension d, the behavior of the determinant changes with this value. 68 4.3.6 Ricci scalar of the parameter manifold. Since the determinant of the QGT is regular (non-zero), it is possible to compute the curvature associated with the system. Keep in mind that the parameter space is two-dimensional, with coordinates λ = (α, J), so the entire geometric information is encoded in the Ricci scalar [19, 36,37]. R = 24t+d−2 Γ [ 4−d 2 ] ( π αL2 )d/2 (t− 2)(t+ d− 2) (4.85) which, we show for our cases of interest: Ricci scalar for the scalar free theory with a source in various (t+ d) dimensions.11 R0+0 = 1 R1+0 = 4 R1+1 = 0 R1+2 = − 16π αL2 R1+3 = − 64π α3/2L3 First, we note that for t = 0, 1 and d = 0, the Ricci scalar is positive, thus the parameter space has a spherical geometry. For any d, with t = 2 we find that the Ricci scalar is zero, which may be interesting for theories of augmented gravity. For t = 1 and d = 1, the Ricci scalar is zero, which tells us that the geometry of the system is planar, meanwhile for t = 1 and 1 < d ≤ 3, the Ricci scalar is negative, but the dependence of the volume ∼ 1 Ld , we can see that it recovers the planar geometry when L→ ∞. It is interesting to note that the Ricci scalar oscillates wiht respect to the growing value of d, where it is considered as a continuous parameter. Moreover, the zeroes are the same when the determinant of the QGT (detQ) is indeterminate, but for the case 1 + 1, where the determinant is finite, but the curvature is zero. Plots of the Ricci scalar R[α, L, t, d]. 3D Plot of the Ricci scalar. Figure 4.11: This 3D plot of the scalar misleads that there are some divergences of the Ricci scalar, when we consider the time and spatial dimensions to be continuous parame- ters of the theory. 11We write Rtd, where t is the time dimension and d the spatial one. 69 Plots of the Ricci scalar R[α, L, t, d]. Plot of the Ricci scalar, varying d. Figure 4.12: Projection of the Ricci scalar, where we fix the mass parameter α, the volume given by L = 3, and show the plots for 0 + 0, 1 + 0, and 1 + d independently, considering d as a continuous parameter. For 1 + d, we can observe that the zeros correspond to the points where the determinant of the QGT diverges, except for the QFT in 1 + 1 dimensions. Plots of the Ricci scalar R[α, L, t, d]. Plot of the Ricci scalar, varying α. Figure 4.13: This plot shows the dependence of α for the Ricci scalar. For 0 + 0, 1 + 0, and 1 + 1, the curvature is not affected by whether the theory becomes massless or not, since they are constant. However, for d > 1, the transition from a massive theory to a massless one results in the divergence of the Ricci curvature. 70 4.4 QGT of two coupled fields in a 0 + 0 QFT. In this example we are going to consider a Lagrangian with two scalar fields ϕ1 and ϕ2 coupled as follows: L = α 2 ( ϕ21 + β2ϕ22 ) + λ 8 ( ϕ41 + 6β2ϕ21ϕ 2 2 + β4ϕ42 ) . (4.86) Here, the parameters of the system are the coupling constants: α, β, and λ. Note that, α can be consider as the mass parameter. It is easy to check that after the transformation ϕ± = 1√ 2 (ϕ1 ± βϕ2), (4.87) the Lagrangian (4.86) becomes L = α 2 ( ϕ2+ + ϕ2− ) + λ 4 ( ϕ4+ + ϕ4− ) . (4.88) Thus, the original Lagrangian can be decoupled into two ϕ4-theories. But we need to take into account the Jacobian of the transformation which is equal to 1+β2 2 , so the normalization of the partition function is correct: Z0 = ∫ dϕ1dϕ2 e −L[ϕ1,ϕ2] = ∫ dϕ1dϕ2 e −[α 2 (ϕ2 1+β 2ϕ2 2)+λ 8 (ϕ4 1+6β2ϕ2 1ϕ 2 2+β 4ϕ4 2)] = 2 1 + β2 ∫ dϕ+dϕ−e −[α 2 (ϕ2 ++ϕ2 −)+λ 4 (ϕ4 ++ϕ4 −)] = 2 1 + β2 ∫ dϕ+dϕ−e −L[ϕ+,ϕ−] (4.89) 4.4.1 Case α > 0: For the calculations of the QGT, we work with the normal modes ϕ±; that is with the uncoupled system. Then, the partition function is given by Z0 = 2 1 + β2 ∫ dϕ+dϕ−e −[α 2 (ϕ2 ++ϕ2 −)+λ 4 (ϕ4 ++ϕ4 −)] (4.90) which can be analytically solved in terms of the modified BesselK function (K): Z0 = αK [ 1 4 , α2 8λ ]2 e α2 4λ 2λ (4.91) It’s interesting to point out that the partition function does not depend on the parameter β. To compute the QGT for this system, we need first to compute the deformation operators Oκ given by Oκ = ∂L ∂κ (4.92) with κ = α, β, λ, being the parameters of the system. Thus, we have the following expressions for each deformation operator: Oα = 1 2 ( ϕ2+ + ϕ2− ) , (4.93) Oβ = α 2β (ϕ+ − ϕ−) 2 + 3λ 8β (ϕ+ + ϕ−) 2 (ϕ+ − ϕ−) 2 + λ 8β (ϕ+ − ϕ−) 4 (4.94) Oλ = 1 4 ( ϕ4+ + ϕ4− ) (4.95) Oβ takes this form because we have to take the derivative w.r.t β in the original Lagrangian (4.86), and then apply the transformationation (4.87). 71 Components of the QGT for α > 0: For completeness, we present the analytical expressions for the QGT components of this system with two coupled fields, as described by the Lagrangian (4.86). The components are expressed in terms of the modified Bessel functions of the first kind (I) and second kind (K). Qαα = 1 8α2λ2K [ 1 4 , α2 8λ ] [ (3α2 + 4λ)K [ 1 4 , α2 8λ ] + α2π2I [ − 1 4 , α2 8λ ] − π2(α2 + 4λ)I [ 1 4 , α2 8λ ] −α2π2 ( I [ 5 4 , α2 8λ ] − I [ 3 4 , α2 8λ ] ) − 4α4K [ 1 4 , α2 8λ ] K [ 3 2 , α2 8λ ] + α4K [ 3 4 , α2 8λ ] ] (4.96) Qαβ = α 2βλ ( K [ 3 4 , α2 8λ ] K [ 1 4 , α2 8λ ] − 1 ) . (4.97) Qαλ = α 32λ3K [ 1 4 , α2 8λ ]2 [ α2π2I [ − 3 4 , α2 8λ ]2 − π2(α2 + 8λ)I [ − 1 4 , α2 8λ ]2 + 2π2(α2 + 8λ)I [ − 1 4 , α2 8λ ] I [ 1 4 , α2 8λ ] − π2(α2 + 8λ)I [ 1 4 , α2 8λ ]2 − 2α2π2I [ − 3 4 , α2 8λ ] I [ 3 4 , α2 8λ ] + α2π2I [ 3 4 , α2 8λ ]2 + 8λK [ 1 4 , α2 8λ ] K [ 3 4 , α2 8λ ] ] . (4.98) Qββ = 1 β2  20 + 5α2 λ + α2K [ 3 4 , α2 8λ ] ( 3K [ 3 4 − 8K [ 1 4 , α2 8λ ] , α 2 8λ ] ) λK [ 1 4 , α2 8λ ]2   . (4.99) Qβλ = 1 2βλ2 [ α2 + 2λ− α2K [ 3 4 , α2 8λ ] K [ 1 4 , α2 8λ ] ] (4.100) Qλλ = 1 64λ4K [ 1 4 , α2 8λ ]2 [ − π2α2I [ − 3 4 , α2 8λ ]2 + π2 ( α4 + 16α2λ+ 16λ2 ) I [ −−1 4 , α2 8λ ]2 − 2π2 ( α4 + 16α2λ+ 16λ2 ) I [ − 1 4 , α2 8λ ] I [ 1 4 , α2 8λ ] + π2 ( α4 + 16α2λ+ 16λ4 ) I [ 1 4 , α2 8λ ]2 + 2π2α4I [ − 3 4 , α2 8λ ] I [ 3 4 , α2 8λ ] − π2α4I [ 4 , α2 8λ ]2 − 24α2λK [ 1 4 , α2 8λ ] K [ 3 4 , α2 8λ ] ] (4.101) 4.4.2 Case for α < 0: It is remarkable that in 0 + 0 dimensions we can compute analytically the QGT when we consider α < 0, which is associated to the mass term for the scalar free theory. In this case the operators are the same, but the partition function changes: Z0< = −π 2α e α2 4λ 4λ ( I [ − 1 4 , α2 8λ ] + I [ 1 4 , α2 8λ ] )2 . (4.102) We present the components of the QFT for completeness as it was done for α > 0. 72 Components of the QGT for α < 0: Let ZI = I [ − 1 4 , α2 8λ ] + I [ 1 4 , α2 8λ ] (4.103) Qαα = − 1 8α2λ2 Z2 I [ (α4 + 4α2λ)I [ − 1 4 , α2 8λ ]2 + 2α2(α2 + 4λ)I [ − 1 4 , α2 8λ ] I [ 1 4 , α2 8λ ] + (α4 + 4α2λ− 16λ2)I [ 1 4 , α2 8λ ]2 − 8α2λI [ 1 4 , α2 8λ ] ( I [ 3 4 , α2 8λ ] + I [ 5 4 , α2 8λ ] ) − α4 ( I [ 3 4 , α2 8λ ] + I [ 5 4 , α2 8λ ] )2 ] (4.104) Qαβ = − 1 2αβλZI [ α2I [ − 1 4 , α2 8λ ] + (α2 + 4λ)I [ 1 4 , α2 8λ ] + α2 ( I [ 3 4 , α2 8λ ] + I [ 5 4 , α2 8λ ] ) ] (4.105) Qαλ = − 1 16αλ3Z2 I [ (α4 + 8α2λ)I [ − 1 4 , α2 8λ ]2 + α2(α2 + 8λ)I [ 1 4 , α2 8λ ]2 + 2I [ − 1 4 , α2 8λ ] ( (α4 + 8α2λ+ 8λ2)I [ 1 4 , α2 8λ ] + 2α2λ ( I [ 3 4 , α2 8λ ] + I [ 5 4 , α2 8λ ] )) − ( α2I [ 3 4 , α2 8λ ] + 4α2λI [ 1 4 , α2 8λ ] )( I [ 3 4 , α2 8λ ] + I [ 5 4 , α2 8λ ] ) ] (4.106) Qββ = 1 8α2β2λZ2 I [ 5(α4 + 4α2λ)I [ − 1 4 , α2 8λ ]2 + (5α4 + 52α2λ+ 48λ2)I [ 1 4 , α2 8λ ]2 + 8α2(α2 + 3λ)I [ 1 4 , α2 8λ ] ( I [ 3 4 , α2 8λ ] + I [ 5 4 , α2 8λ ] ) + 3α4 ( I [ 3 4 , α2 8λ ] + I [ 5 4 , α2 8λ ] )2 + 2I [ − 1 4 , α2 8λ ] { (5α4 + 36α2λI [ 1 4 , α2 8λ ] + 4α4 ( I [ 3 4 , α2 8λ ] + I [ 5 4 , α2 8λ ] )} ] (4.107) Qβλ = 1 βλ2ZI [ (α2 + 2λ)I [ − 1 4 , α2 8λ ] + (α2 + 6λ)I [ 1 4 , α2 8λ ] + α2 ( I [ 3 4 , α2 8λ ] + I [ 5 4 , α2 8λ ] ) ] (4.108) Qλλ = 1 32λ4ZI [ (α4 + 16α2λ+ 16λ2)I [ − 1 4 , α2 8λ ]2 + (α4 + 16α2λ+ 48λ2)I [ 1 4 , α2 8λ ]2 4α2λI [ 1 4 , α2 8λ ] ( I [ 3 4 , α2 8λ ] + I [ 5 4 , α2 8λ ] ) − α4 ( I [ 3 4 , α2 8λ ] + I [ 5 4 , α2 8λ ] )2 2I [ − 1 4 , α2 8λ ] { (α4 + 15α2λ+ 40λ2)I [ 1 4 , α2 8λ ] + 6α2λ ( I [ 3 4 , α2 8λ ] + I [ 5 4 , α2 8λ ] )} ] (4.109) 73 Plots of the Quantum Geometric Components for α > 0 and α < 0: As is evident from the equations for the components of the Quantum Geometric Tensor (QGT), regardless of whether α takes positive or negative values, they exhibit a strong dependence on the behavior of the modified Bessel functions K and I, and on the relation between the parameters α and λ through the ratio α2 8λ , which appears as the argument of these functions. In the following figures, we display in yellow the plots corresponding to the QGT components for α < 0, and in red those for α > 0. From these plots, we observe that for the components Qαα, Qαλ, Qββ , and Qβλ, fixing the parameter α to a constant α0 such that α0 > 0 and comparing with −α0 < 0, the plot corresponding to the negative value of α (yellow) lies above the plot associated with the positive values of α (red). For Qαβ and Qβλ, this behavior holds for positive values of β, but reverses for negative β. Regardless of the sign of α, all components diverge as λ approaches zero. Another notable observation is that as β approaches zero, the components Qiβ , where i = α, β, or λ, also diverge. In contrast, the remaining components exhibit a smooth transition between positive and negative values of β. This behavior persists when we fix the value of λ instead of α. Continuing the analysis with λ fixed, we find that for negative values of α, the plots grow unboundedly, while for positive α, the plots decay smoothly to zero. These patterns stem from the asymptotic behavior of the modified Bessel functions: the I function (governing the negative α case) diverges, while the K function (governing the positive α case) vanishes. Finally, when keeping β constant, the behavior of the QGT components between positive and negative regions of α remains qualitatively similar. They all diverge as λ → 0, while the transition across the sign of α is smooth. In summary, the behavior of the QGT components reflects what is expected from the structure of the Lagrangian: discontinuities or divergences as α, λ, or β approach zero, indicating potential quantum phase transitions. 3D plots for Qαα[α, β, λ]. Figure 4.14: Qαα[±α0, β, λ]. Figure 4.15: Qαα[α, β0, λ]. Figure 4.16: Qαα[α, β, λ0]. 3D Plots Qαβ [α, β, λ]. Figure 4.17: Qαβ [±α0, β, λ]. Figure 4.18: Qαβ [α, β0, λ]. Figure 4.19: Qαβ [α, β, λ0]. 74 Qαλ[α, β, λ]. Figure 4.20: Qαλ[±α0, β, λ]. Figure 4.21: Qαλ[α, β0, λ]. Figure 4.22: Qαλ[α, β, λ0]. 3D plots Qββ [α, β, λ]. Figure 4.23: Qββ [±α0, β, λ]. Figure 4.24: Qββ [α, β0, λ]. Figure 4.25: Qββ [α, β, λ0]. 3D plots Qβλ[α, β, λ]. Figure 4.26: Qβλ[±α0, β, λ]. Figure 4.27: Qβλ[α, β0, λ]. Figure 4.28: Qβλ[α, β, λ0]. 3D plots Qλλ[α, β, λ]. Figure 4.29: Qλλ[±α0, β, λ]. Figure 4.30: Qλλ[α, β0, λ]. Figure 4.31: Qλλ[α, β, λ0]. 75 Determinant of the QGT for α > 0 and α < 0. 3D Plots detQ[α, β, λ]. Figure 4.32: detQ[±α0, β, λ]. Figure 4.33: detQ[α, β0, λ]. Figure 4.34: detQ[α, β, λ0]. Analogous to the components of the QGT, the determinant exhibits a discontinuity between the plots for fixed values of α = ±α0, where the plot for negative α (yellow) lies above the plot for positive α (red). When β and λ are held fixed, the transition between the negative (yellow) and positive (red) regions of α is smooth. The divergences occur when λ→ 0 for a constant β and when β → 0 for a constant λ. This behavior of the determinant reinforces the conclusions drawn from the QGT components: when α, β, or λ approach zero, a quantum phase transition may occur: massless theory (α→ 0), absence of quadratic coupling between scalar fields (β → 0), and the disappearance of the quartic potential for the theory (λ→ 0). 76 Chapter 5 Conclusions The Quantum Geometric Tensor (QGT) encapsulates two fundamental aspects of a quantum system: 1. Quantum Metric Tensor (QMT): This component of the QGT measures the inifinesimal distance between quantum states as the parameters of the system vary. It quantifies the “closeness” of the quantum states in the parameter space, providing insight into how small changes in parameters affect the state. The QMT is directly related to the quantum susceptibility, which is crucial in the understanding of quantum phase transitions (QPT’s) and the purity of the system. A large value of the QMT implies that the quantum states are highly sensitive to the changes in the parameters, indicating potential transitions between different quantum phases. 2. Berry curvature: This component captures the topological features of the system and describes the geometric structure of the parameter space. It can be interpreted as the geometric analogue of the electro- magnetic field. It plays a crucial role in understanding QPT’s, geometric phases, interference, and other topological aspects of the systems, as an example, the appearance of nontrivial phases or quantum anomalies in the Aharonov-Bohm effect or topological insulators (e.g. Weyl semimetals). The geometric phases are acquired by the quantum states after undergoing adiabatic changes in the parameters of the system. In this text, we show an extension of the QGT to systems where are embedded in curved configuration spaces. This gives rise to extra terms for the QGT, the QMT and the Berry curvature, which are purely geometric corrections governed by what we defined as the deformation vector, σρ. Moreover, the Berry connection is modified by this vector, and we had to redefine it in such manner that it mantains the reality condition as in the usual (“flat”) case [15]. The examples given come from some holomorphic gauge symmetries in the complex plane, and to be analysed we had to use the Laplace-Beltrami equation. Also, it is important that the systems come from a classical Lagrangian, and when we quantized the system is important to symmetrize it, such that the order of the operators qi and pi do not give rise to quantum anomalies. We further examined the QGT for spin- 12 particles and antiparticles using two different representations of the γ matrices: the Dirac and Weyl representations. The motivation behind this was to investigate whether it might be possible to distinguish Dirac from Weyl particles through geometric means, potentially allowing experimentalists to probe this distinction by varying physical parameters. However, our analysis showed that the QGT is the same in both representations, which, although disappointing for this purpose, aligns with our expectations. Then, we used the Dirac representation to compute the QGT focusing on the effects of mass and momen- tum along the p3-axis. By considering the full solutions of the Dirac equation and regularizing the integration over space, we derived the components of the QGT, which are the same as those found in previous sections when accounting for the dispersion relation. The components of the QGT depend on the mass m, the mo- mentum p3 and the relativistic relation. Moreover, the determinant of the QGT shows an interesting feature: when the momentum components p1 and p2 are zero, the determinant of the QGT vanishes. This corresponds to the case where the dispersion relation p20 = m2 + p23 holds, indicating a special case in which the QGT degenerates. We explore the fidelity approach for both cases, where only the mass m is varied, and when both the mass m and the p3-momentum vary. One can recover the first case when setting p3 constant and applying the 77 dispersion relation. Thus, we focus on the analysis of the second one. This measure of quantum state overlap between different values of mass and momentum shows a strong dependence on the momentum component p3. The quantum fidelity decreses as the parameters deviate from the reference values, especially for negative momentum values, highlighting the sensitivity of the system to changes in p3. That is, the QGT is a good approximation to the quantum susceptibility only in certain regions. For the Weyl semimetal analysed in the thesis, we chose the simplest Hamiltonian and determined the eigenvctors and eigenvalues of the system. The primary goal was to investigate the geometric properties of the Weyl semimetal by calculating the QMT componentes, the Berry curvature and the Ricci scalar of the parameter manifold through the QMT. For this system, we considered the momenta ki, i = 1, 2, 3 to be the parameters of the system. The QMT describe the local geometry of the parameter space in terms of the momentum variables. The non-zero Berry curvature was found, and it was shown to match the known form from condesed matter theory. The Ricci scalar of the parameter manifold was computed, yielding a positive constant, resulting in a spherical geometry of radius r = 1√ 2 , allowing the geometric interpretation for the Weyl semimetal. That is the parameter space (the k-space) was shown to resemble a 2-dimensional sphere, which is closely related to the Bloch sphere used to describe the state space of quantum systems. This simple case study highlights the importance of the QGT in understanding the geometric and topolog- ical properties of systems like Weyl semimetals. The results not only confirm the known topological features but also provide a new perspective on the underlying geometry of quantum states in these materials. Finally, we computed the QGT in the context of quantum field theories (QFTs), focusing in particular on the 0 + 0-dimensional case. Our goal was to explore whether this simplest QFT setting can offer insights or infer properties relevant to more general quantum field theories. To explore the extension of the QGT from quantum mechanics to QFT we used the path integral formal- ism, which provides a natural framework to describe quantum fields and their dynamics, and it allows for the computation of the QGT in a direct manner. A central difficulty in this formalism lies in the absence of a Berry connection, i.e., there is no known gauge potential whose exterior derivative yields the Berry curvature, as occurs in the usual (representation) formalism in quantum mechanics. Despite this limitation, the full QGT can still be constructed in terms of two-point correlation functions of the operators Oα = ∂L ∂λα , where L is the Lagrangian of the system. These operators are associated with infinitesimal deformations of the theory along directions in the parameter space M. For the 0 + 0 dimensional case, where the QFT reduces to a real random variable, we are able to compute explictly the vacuum overlap and obtain exact, closed-form expressions for the QGT in this setting. Thus, giving a concrete example of how geometric information is encoded in the overlap of the ground states under parameter variations, even in the absence of time or spacial dimensions. First, we consider the theory in presence of a source term Jϕ, but setting J to be a parameter of the system, i.e. a constant different from zero as is usually treated in QFT. By doing so, we generalized the components of the QGT for arbitrary time (t) and spatial (d) dimensions, obtaining explicit expressions that reveal how the tensor depends on the physical parameters of the theory: the mass term α, the external parameter J , and the system size L. We observe that all components diverge as α→ 0, signaling the presence of a phase transition, consistent with the massless limit of the theory. Additionally, we obtained a general expression for the determinant of the QGT, which interestingly does not depend on J but instead scales with L2d and includes the same Γ [ 4−d 2 ] factor, implying the same divergence pattern for even spatial dimensions. The determinant also exhibits an intricate dependence on α, scaling as α−r for most cases of interest, though it could potentially become proportional to a positive power of α when d > 6t. These results collectively highlight the rich geometric structure of the theory across dimensions and provide insight into its critical behavior, while also signaling the need for dimensional regularization in higher-dimensional limits where the theory becomes ill-defined. The Ricci scalar of the parameter manifold was also computed. It captures key geometric features of the theory across dimensions. It is positive or zero in low-dimensional cases, vanishes for t = 2, and becomes negative or divergent in higher dimensions, especially in the massless limit. Interestingly, its zeros align with the divergences of the QGT determinant, except in the (1 + 1) case, highlighting a rich interplay between geometry and quantum criticality. 78 The last example is a 0 + 0 dimensional QFT with two coupled scalar fields. Here, we analyzed the QGT structure arising of a Lagrangian with three parameters, which one of it disappears through a linear transformation, in the normal modes. It is interesting to note that through the operators Oα, the QGT is sensible to reparametrizations and interactions, even when the free energy remains unaffected. The analysis also distinguishes between the cases α > 0 and α < 0, revealing qualitatively different expressions for the QGT components due to the analytic structure of the partition function in each regime. What further directions could we explore? For instance, in momentum space, we might introduce a momentum-dependent metric, thereby encoding curvature directly into that space. This could lead to a notion of curvature for fermions. Additionally, the results obtained in 0 + 0 dimensions could potentially be generalized to higher dimensions with appropriate couplings. Can the quantum mechanical treatment of fermions be extended to quantum fields? In that case, the path integral formulation with Dirac fermions would involve an infinite number of degrees of freedom. For curved spaces, one could investigate how to extend the formalism to finite-dimensional Hilbert spaces. Moreover, the 0+0 dimensional case might provide predictive insights into entanglement in quantum systems. 79 80 Appendix A Complex and Holomorphic Gauge Symmetries. This appendix is intended to complement the chapter on the curved quantum geometric tensor (QGT), where most of the examples discussed are based on modifications of the standard harmonic oscillator. To provide a clearer understanding of its quantization in these contexts, we present here a formulation that extends the system to the complex plane. It’s possible to extend any system to the complex plane with the only condition that these extensions are given only in terms of holomorphic variables. [38]. This condition implies that there exists a gauge symmetry hidden in the theory that allows to project the extended complex system in different ways to the real physical space. It’s important to notice that this gauge symmetry not necessarily is canonical or unitary. By requiring the Lagrangian to be holomorphic, the Cauchy-Riemann conditions are automatically sat- isfied, and the resulting primary constraints become the generators of the gauge transformations associated with this symmetry. Let’s consider a complex-holomorphic Lagrangian: L(z, ż) = gab(z) 2 żażb − V (z) (A.1) with a = 1, ..., n. We can write za = xa + iya with 2n real variables. This Lagrangian is invariant under the coordinate transformation: x ′a = xa + λa(t) & y ′a = ya + iλa(t) (A.2) where λa(t) are arbitrary functions that depend only on the time (t). This means, that we have a local symmetry. The variations of the coordinates and the Lagrangian with respect to this transformation are δza = z ′a − za = 0 & δL = 0 (A.3) thus, it is a legitimate symmetry. The canonical conjugate momenta to x and y are pxa = gabż b pya = igabż b (A.4) respectively. Thus, we obtain n primary constrictions: Φa = pxa + ipya ∼ 0 (A.5) then, making use of these constrictions, our canonical Hamiltonian is HC = gab 2 pxa pxb + V (x) (A.6) 81 such that, the total Hamiltonian of the system must be HT = HC + µaΦa (A.7) where µa are n Lagrangian multipliers. It is easy to check that the symplectic structure of this system is given by the Poisson brackets: {xa, pxb } = δab = {ya, pyb} (A.8) then, the temporal evolution of our constrictions is Φ̇a = {Φa, Ht} = 0. (A.9) This, tells us that our primary constrictions are of the first class too. Therefore, they are the generators of the gauge symmetries given by δxa = {xa, ϵbΦb} = ϵa & δya = {ya, ϵbΦb} = iϵb. (A.10) Thus, it is necessary to include the gauge conditions to remove the extra degrees of freedom of the theory, i.e., Γa(x, y, px, py) ∼ 0. (A.11) Remark: To make the system free of arbitrary parameters, the gauge conditions must imply that the set of constrictions χA = (Γa,Φa) generates a set of second class constrictions, this means, that det ( {χA, χB} ) ̸= 0 (A.12) must be satisfied. A.1 Complex Harmonic Oscillator. Let’s start with the action of the harmonic oscillator in the complex plane parametrized as follows: S = τ1 ∫ τ0 dτ ( ż2 2ṫ − ω2 2 ṫz2 ) (A.13) which gives the following two constrictions of first class Φ = px + ipy ∼ 0 (A.14) Ψ = pt + p2x 2 + ω2 2 (x+ iy)2 ∼ 0. (A.15) To quantize this system, we fix partially the gauge using one gauge condition for Φ. Meanwhile, for Ψ we use the Dirac’s quantization condition: Ψ̂|ψ⟩phys = 0. (A.16) Then, we can fix the gauge to γ0 = y ∼ 0 (A.17) such that the Dirac bracket1 {x, px}∗ = 1 (A.18) same as the usual harmonic oscillator. Otherwise, we can use the gauge condition γ1 = y − i(x− U1/2(x)) ∼ 0 (A.19) where we get that the constriction Ψ becomes Ψ = ( pt + p2x 2 + ω2 2 U(x) ) (A.20) with Dirac bracket {x.px}∗ = 2U1/2 ∂xU . (A.21) 1The Dirac bracket are a generalization of the Poisson brackets for system with constrictions. 82 A.2 Calculation of the Dirac brackets. First, let’s consider the constrictions Φ and Ψ, that at this moment are of first class, i.e., {Φ,Ψ} = {px + ipy, pt + p2x 2 + ω2 2 (x+ iy)2} = ω2(x+ iy)(1 + i2) = 0. (A.22) Secondly, we will consider the Dirac quantization (A.16) condition for Ψ. Then, we introduce the gauge condition γ to fix the degree of freedom for y. This makes Φ and γ constrictions of the second class. Moreover, it is necessary to compute the Dirac bracket to fix the degree of freedom that we are missing, in this case, x. In general, the Dirac bracket is defined as {f, g}∗ = {f, g}PB − ∑ a,b {f, χa}Cab{Xb, g} (A.23) where χa are second class constrictions and Cab is the inverse matrix of Cab = {Xa, Xb}PP . (A.24) In our case, we want to calculate the Dirac bracket {x, px}∗. To do so, first, let’s calculate the Poisson bracket between the constrictions Φ and γ1: {Φ, γ1} = {px + ipy, y − i(x− U1/2(x)} = i− i 2 U1/2∂xU − i = − i 2 U1/2∂xU (A.25) which is the matrix element of CΦγ1 . Then, the Dirac bracket {x, px}∗ = {x, px} − {x,Φ}Cϕγ1{γ1, px} = 1 − (1) ( 2U1/2 i∂xU )( −i+ i 2 U1/2∂xU ) = 2U1/2 ∂xU . (A.26) Then, we cannot realised the momentum as usual: px → −iℏ∂x, but we must realise it in a way that satisfies the Dirac bracket adequately. Thus, the conjugate momentum in our case is realised as px → −iℏ2U1/2 ∂xU ∂x . 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