Hilbert space geometry and quantum chaos

TL;DR

This paper explores the geometry of quantum states via the quantum geometric tensor, revealing how the shape of parameter space indicates ergodic or integrable behavior in complex quantum systems.

Contribution

It analyzes the quantum Riemannian metric derived from the QGT for random matrix Hamiltonians, linking geometric features to ergodic and integrable phases.

Findings

Ergodic phase corresponds to smooth manifold geometry.

Integrable limit shows singular geometry with conical defect.

Parameter space geometry reflects ergodic-nonergodic transition.

Abstract

The quantum geometric tensor (QGT) characterizes the Hilbert space geometry of the eigenstates of a parameter-dependent Hamiltonian. In recent years, the QGT and related quantities have found extensive theoretical and experimental utility, in particular for quantifying quantum phase transitions both at and out of equilibrium. Here we consider the symmetric part (quantum Riemannian metric) of the QGT for various multi-parametric random matrix Hamiltonians and discuss the possible indication of ergodic or integrable behaviour. We found for a two-dimensional parameter space that, while the ergodic phase corresponds to the smooth manifold, the integrable limit marks itself as a singular geometry with a conical defect. Our study thus provides more support for the idea that the landscape of the parameter space yields information on the ergodic-nonergodic transition in complex quantum systems, including the intermediate phase.