Geometry of the parameter space of a quantum system: Classical point of view

TL;DR

This paper introduces a new semiclassical approach to derive classical analogs of the quantum metric tensor and Berry curvature, applicable to various classical systems derived from quantum models, with validation on multiple examples.

Contribution

It presents the first method to relate quantum geometric objects to their classical counterparts using semiclassical approximation in the Lagrangian formalism.

Findings

Validated approach on five classical systems

Established relation between quantum and classical metric tensors

Applicable to systems with bosonic and fermionic degrees of freedom

Abstract

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. We consider classical integrable systems and report a new approach 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. Our approach arises from the semiclassical approximation of the Berry curvature and the quantum metric tensor in the Lagrangian formalism. We also exploit this semiclassical approximation to establish, for the first time, the relation between the quantum metric tensor and its classical counterpart. We illustrate and validate our approach 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, we mention some potential applications of this approach and possible generalizations that can be of interest in the field of condensed matter physics