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

Alvarez‐Jimenez, Javier; González, Diego López; Gutiérrez‐Ruiz, Daniel; Vergara, J. David

doi:10.1002/andp.201900215

Cited by 10 publications

(19 citation statements)

References 64 publications

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“…(100) vanishes, and then both metrics produce the same parameter structure, satisfying the semiclassical relation established in Ref. [37]. The origin of the extra term can be attributed to the fact that the quantum metric tensor (90) has corrections of order 2 to the classical metric (99).…”

Section: Classical Analog Of the Quantum Metric Tensorsupporting

confidence: 71%

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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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Section: Classical Analog Of the Quantum Metric Tensorsupporting

confidence: 71%

“…[38,39,40], for instance), little research has focused on its parameter space [37]. Moreover, its associated quantum metric tensor is known only for the ground state [37]. Here, as a direct application of Eq.…”

Section: Example: Linearly Coupled Harmonic Oscillatorsmentioning

confidence: 99%

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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“…This volume contains papers investigating properties of Dirac materials both in-and out-of-equilibrium. J. Alvarez-Jimenez et al reports, in their original article, [4] a new approach to obtain the classical analogs of the quantum metric tensor and the Berry curvature. They show how this approach is advantageous over other procedures for classical systems that correspond to quantum systems with bosonic and fermionic degrees of freedom.…”

Section: Doi: 101002/andp202000037mentioning

confidence: 99%

Dynamic Quantum Matter

et al. 2020

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We are witnessing rapid developments in the field of quantum materials with the focus on some of the most profound concepts in condensed matter, including entangled orders, quantum coherence, and quantum topology. Quantum correlations in these materials naturally reveal themselves in the time domain, since their temporal evolution is governed by the full Hamiltonian, which contains multiple interactions. Therefore, nonequilibrium quantum dynamics emerges as a design principle to create desired quantum materials and functionalities. We see a growing focus on dynamics as a way to understand and control the fundamental physical processes that emerge due to quantum coherence of entangled quantum matter. Commensurate with the developments in theory and modeling of quantum materials, we see a rapid rise of new probes for the examination of quantum matter. These new probes include ultra-fast optics, free-electron lasers, and new neutron scattering facilities with capabilities to probe quantum matter at the increasingly short time scale while maintaining high-spatial resolution. Reflecting on recent progress and excitement in the field, we organized a workshop on Dynamic Quantum Matter Dec 10-14, 2018 at the Nordic Institute for Theoretical Physics (NORDITA, https://indico.fysik.su.se/event/6504/). As a result of this conference, we put together a special issue of the Annalen der Physik on "Dynamic Quantum Matter". We summarize the contributions in this volume. This special issue contains a rapid research letter, four feature articles, five original papers, and a perspective on the dynamics of various forms of quantum matter and materials ranging from superconductivity to magnetism, bosonic and fermionic Dirac materials ferro-states in 2D materials.

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“…In Ref. [22], it was proved that the classical metric results from a semiclassical approximation of the QMT under the time-dependent Lagrangian approach introduced in [23]. The classical metric possesses the same properties as its quantum counterpart: it is positive semidefinite, gauge invariant, and it transforms as a rank-two covariant tensor.…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, the classical metric shows its relevance emerging as a tool that, through purely classical functions and a classical torus average, provides a result consistent with the quantum description in many cases. We must mention, however, that some differences between the classical and quantum metrics may appear essentially due to operator-ordering ambiguities, which may result in (i) anomalies that contribute with additional terms [22,25], (ii) differences coming from the fact that there might be distinct quantizations for a given classical Hamiltonian.…”

Section: Introductionmentioning

confidence: 99%

Classical description of the parameter space geometry in the Dicke and Lipkin-Meshkov-Glick models

Gonzalez,

Gutiérrez-Ruiz,

Vergara

2021

Preprint

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We study the classical analog of the quantum metric tensor and its scalar curvature for two well-known quantum physics models. First, we analyze the geometry of the parameter space for the Dicke model with the aid of the classical and quantum metrics and find that, in the thermodynamic limit, they have the same divergent behavior near the quantum phase transition, as opposed to their corresponding scalar curvatures which are not divergent there. On the contrary, under resonance conditions, both scalar curvatures exhibit a divergence at the critical point. Second, we present the classical and quantum metrics for the Lipkin-Meshkov-Glick model in the thermodynamic limit and find a perfect agreement between them. We also show that the scalar curvature is only defined on one of the system's phases and that it approaches a negative constant value. Finally, we carry out a numerical analysis for the system's finite sizes, which clearly shows the precursors of the quantum phase transition in the metric and its scalar curvature and allows their characterization as functions of the parameters and of the system's size.

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

Cited by 10 publications

References 64 publications

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

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

Dynamic Quantum Matter

Classical description of the parameter space geometry in the Dicke and Lipkin-Meshkov-Glick models

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