The quantum geometric tensor from generating functions

Alvarez‐Jimenez, Javier; Vergara, J. David

doi:10.1142/s0219749919500175

Cited by 7 publications

(12 citation statements)

References 26 publications

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“…By comparing (33) and (34), and taking into account the Bohr-Sommerfeld quantization rule for action variable I = ℏ∕2, it is direct to check that these curvatures satisfy the relation (19). Hence, in this example, besides showing the applicability of Equations (18) and (20), we have verified that these expressions yield the expected results for the classical metric and curvature.…”

Section: Generalized Harmonic Oscillatorsupporting

confidence: 62%

“…We will see this through examples. One way to anticipate the presence of such quantum anomalies would be the appearance of loop diagrams in the computation of Equation . Nevertheless, this issue requires further investigation and is beyond the scope of the present study.…”

Section: Classical Counterparts Of the Quantum Metric Tensor And The mentioning

confidence: 91%

“…[24]. Note that while relation (19) for the curvatures involves the factor 1∕ℏ, the analogous relation (17) for the metrics involves the factor 1∕ℏ 2 . The origin of these different factors can be traced back to the fact that in the former case the replacement of the commutators by the Poisson brackets introduces ℏ, while in the latter case the corresponding replacement of the anticommutators does not.…”

Section: Classical Counterparts Of the Quantum Metric Tensor And The mentioning

confidence: 99%

“…Actually, this feature will appear in all our illustrative examples. The reason is that the computation of the expectation values in the quantum metric naturally incorporates loop diagrams, [19] and since that is a purely quantum effect, we must postulate a different quantization rule for superior powers of I. In any case, if this empirical quantization rule were not taken into account, the quantum and classical metrics would only differ by a global numeric factor, although their parameter dependence would be identical.…”

Section: Generalized Harmonic Oscillatormentioning

confidence: 99%

“…In contrast to the standard Hamiltonian approach, the Lagrangian approach involves perturbations of the Lagrangian in the parameter space rather than perturbations of quantum states, which makes it suitable not only for quantum field theories but also for systems where the exact solution is not available. [19] On the other hand, in the context of thermodynamic systems, the classical counterparts of the quantum metric tensor were introduced in the seminal works of Weinhold [20] and Ruppeiner, [21] and were subsequently implemented in the broader frame known as geometrothermodynamics. [22] In the framework of classical integrable systems (those for which it is possible to introduce action-angle variables), both the Berry curvature and the quantum metric tensor have a counterpart.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Geometry of the Parameter Space of a Quantum System: Classical Point of View

Alvarez‐Jimenez¹,

González²,

Gutiérrez‐Ruiz³

et al. 2019

Annalen der Physik

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The local geometry of the parameter space of a quantum system is described by the quantum metric tensor and the Berry curvature, which are two fundamental objects that play a crucial role in understanding geometrical aspects of condensed matter physics. Classical integrable systems are considered and a new approach is reported to obtain the classical analogs of the quantum metric tensor and the Berry curvature. An advantage of this approach is that it can be applied to a wide variety of classical systems corresponding to quantum systems with bosonic and fermionic degrees of freedom. The approach used arises from the semiclassical approximation of the Berry curvature and the quantum metric tensor in the Lagrangian formalism. This semiclassical approximation is exploited to establish, for the first time, the relation between the quantum metric tensor and its classical counterpart. The approach described is illustrated and validated by applying it to five systems: the generalized harmonic oscillator, the symmetric and linearly coupled harmonic oscillators, the singular Euclidean oscillator, and a spin‐half particle in a magnetic field. Finally, some potential applications of this approach and possible generalizations that can be of interest in the field of condensed matter physics are mentioned.

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Section: Generalized Harmonic Oscillatorsupporting

confidence: 62%

Section: Classical Counterparts Of the Quantum Metric Tensor And The mentioning

confidence: 91%

Section: Classical Counterparts Of the Quantum Metric Tensor And The mentioning

confidence: 99%

Section: Generalized Harmonic Oscillatormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Geometry of the Parameter Space of a Quantum System: Classical Point of View

Alvarez‐Jimenez¹,

González²,

Gutiérrez‐Ruiz³

et al. 2019

Annalen der Physik

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Generalized quantum geometric tensor for excited states using the path integral approach

Juárez,

Gonzalez,

Gutiérrez-Ruiz

et al. 2023

Phys. Scr.

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The quantum geometric tensor, composed of the quantum metric tensor and Berry curvature, fully encodes the parameter space geometry of a physical system. We ﬁrst provide a formulation of the quantum geometrical tensor in the path integral formalism that can handle both the ground and excited states, making it useful to characterize excited state quantum phase transitions (ESQPT). In this setting, we also generalize the quantum geometric tensor to incorporate variations of the system parameters and the phase-space coordinates. This gives rise to an alternative approach to the quantum covariance matrix, from which we can get information about the quantum entanglement of Gaussian states through tools such as purity and von Neumann entropy. Second, we demonstrate the equivalence between the formulation of the quantum geometric tensor in the path integral formalism and other existing methods. Furthermore, we explore the geometric properties of the generalized quantum metric tensor in depth by calculating the Ricci tensor and scalar curvature for several quantum systems, providing insight into this geometric information.

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Phase space formulation of the Abelian and non-Abelian quantum geometric tensor

González¹,

Gutiérrez‐Ruiz²,

Vergara³

2020

J. Phys. A: Math. Theor.

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The geometry of the parameter space is encoded by the quantum geometric tensor, which captures fundamental information about quantum states and contains both the quantum metric tensor and the curvature of the Berry connection. We present a formulation of the Berry connection and the quantum geometric tensor in the framework of the phase space or Wigner function formalism. This formulation is obtained through the direct application of the Weyl correspondence to the geometric structure under consideration. In particular, we show that the quantum metric tensor can be computed using only the Wigner functions, which opens an alternative way to experimentally measure the components of this tensor. We also address the non-Abelian generalization and obtain the phase space formulation of the Wilczek-Zee connection and the non-Abelian quantum geometric tensor. In this case, the non-Abelian quantum metric tensor involves only the non-diagonal Wigner functions. Then, we verify our approach with examples and apply it to a system of N coupled harmonic oscillators, showing that the associated Berry connection vanishes and obtaining the analytic expression for the quantum metric tensor. Our results indicate that the developed approach is well adapted to study the parameter space associated with quantum many-body systems.

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The quantum geometric tensor from generating functions

Cited by 7 publications

References 26 publications

Geometry of the Parameter Space of a Quantum System: Classical Point of View

Geometry of the Parameter Space of a Quantum System: Classical Point of View

Generalized quantum geometric tensor for excited states using the path integral approach

Phase space formulation of the Abelian and non-Abelian quantum geometric tensor

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