Geometry of quantum states and chaos-integrability transition

TL;DR

This paper explores the geometric structure of quantum states in random matrix Hamiltonians, analyzing the transition from integrability to chaos through metrics like fidelity susceptibility and quantum metric tensor.

Contribution

It introduces a geometric framework for understanding quantum state transitions in random matrix ensembles, linking correlation functions to fidelity susceptibility and solving geodesic equations.

Findings

Fidelity susceptibility relates to spectral form factors and energy level correlations.

Geodesic distances indicate finite reachability from integrable to chaotic phases.

Fidelity susceptibility exhibits universal features across different random matrix ensembles.

Abstract

We consider the geometry of quantum states associated with different classes of random matrix Hamiltonians, in particular ensembles that show integrability to chaotic transition in terms of the nearest neighbour energy level spacing distribution. In the case that the total Hamiltonian contains a single parameter, the distance between two states is captured by the fidelity susceptibility, whereas, when the total Hamiltonian contains multiple parameters, this distance is given in terms of the quantum metric tensor. Since the fidelity susceptibility is closely related to the two-point correlation function, we first calculate the relevant correlation functions of a random matrix belonging to the Gaussian unitary ensemble in terms of the spectral form factor of the total Hamiltonian, show how to obtain the fidelity susceptibility from this correlation function, and explain the role played by energy level correlation. Next, by performing suitable coordinate transformations, we solve the geodesic equations corresponding to the quantum metric tensor obtained from an integrability-breaking random matrix Hamiltonian and obtain the geodesic distance between two points on the parameter manifold to show that any point far away from the integrable phase can be reached by a finite value of this distance. Finally, we obtain and discuss different properties of the fidelity susceptibility associated with Hamiltonians belonging to another random matrix ensemble which shows integrability to chaos transition, namely the Gaussian $\beta$-ensembles with general values of the Dyson index $\beta$, and show that the fidelity susceptibility shares generic features with the first class of Hamiltonians.