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

Abstract

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.

Extracting non-Abelian quantum metric tensor and its related Chern numbers

“…Note that the formulas in Eqs. (5)(6)(7)(8)(9) naturally recover to those for the non-degenerate case when N = 1. Even in the degenerate case, the distance dS 2 also corresponds to the transition probability from the degenerate ground state |Ψ 0 (λ) to excited states after a sudden quench of the parameter from λ to λ+dλ.…”

“…For degenerate quantum states, the complete geometry is characterized by the non-Abelian QGT [2,4,5,[7][8][9]. When the degenerate states are N -fold, the non-Abelian QGTs are naturally defined on the U (N ) vector bundle and contain the non-Abelian generalizations of the quantum metric and Berry curvature.…”

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

“…A different perspective in the study of quantum systems can also be addressed. We could use the Wigner function formulation of the QMT [56] to study a variety of manybody systems and see whether it provides a deeper insight not only into the QPTs, which refer to the ground state, but also into excited-state quantum phase transitions (ESQPTs) [57]. Additionally, the Wigner function formalism may shed some light on the semiclassical approximation and help clarify the anomalies' role that accounts for the difference between the classical and quantum results.…”

Quantum geometric tensor and quantum phase transitions in the Lipkin-Meshkov-Glick model

“…This quantity has an abrupt change in a quantum phase transition, and it has been used extensively (see Refs. [11][12][13] and references therein). Moreover, alternative definitions of the fidelity and machine learning approaches have recently been used to study phase transitions on spin models with Hamiltonians depending on parameters [14].…”