Hilbert-space geometry of random-matrix eigenstates

TL;DR

This paper analytically derives the probability distribution of the quantum geometric tensor for eigenstates of parameter-dependent random matrices, revealing insights into the geometry of quantum states relevant for various quantum phenomena.

Contribution

It provides the exact joint distribution of the Fubini-Study metric and Berry curvature for the Gaussian Unitary Ensemble, linking quantum geometry with random matrix theory.

Findings

Exact joint distribution of quantum geometric tensor components.

Connections to Levy stable distributions.

Validation through numerical simulations.

Abstract

The geometry of multi-parameter families of quantum states is important in numerous contexts, including adiabatic or nonadiabatic quantum dynamics, quantum quenches, and the characterization of quantum critical points. Here, we discuss the Hilbert-space geometry of eigenstates of parameter-dependent random-matrix ensembles, deriving the full probability distribution of the quantum geometric tensor for the Gaussian Unitary Ensemble. Our analytical results give the exact joint distribution function of the Fubini-Study metric and the Berry curvature. We discuss relations to Levy stable distributions and compare our results to numerical simulations of random-matrix ensembles as well as electrons in a random magnetic field.