Universal distributions of overlaps from generic dynamics in quantum many-body systems

TL;DR

This paper demonstrates that the distribution of overlaps in quantum many-body chaotic systems converges to a universal form, dependent only on dimensionality and boundary conditions, after logarithmic time scaling, supported by analytical and numerical evidence.

Contribution

It introduces a universal overlap distribution for quantum chaotic dynamics, generalizing Porter-Thomas, and connects it to random matrix theory and random quantum circuit models.

Findings

Overlap distribution converges to a universal form in the thermodynamic limit.

Distribution depends only on spatial dimensionality and boundary conditions.

Numerical simulations confirm the universality of the predicted distribution.

Abstract

We study the distribution of overlaps with the computational basis of a quantum state generated under generic quantum many-body chaotic dynamics, without conserved quantities, for a finite time $t$. We argue that, scaling time logarithmically with the system size $t \propto \log L$, the overlap distribution converges to a universal form in the thermodynamic limit, forming a one-parameter family that generalizes the celebrated Porter-Thomas distribution. The form of the overlap distribution only depends on the spatial dimensionality and, remarkably, on the boundary conditions. This picture is justified in general by a mapping to Ginibre ensemble of random matrices and corroborated by the exact solution of a random quantum circuit. Our results derive from an analysis of arbitrary overlap moments, enabling the reconstruction of the distribution. Our predictions also apply to Floquet circuits, i.e., in the presence of mild quenched disorder. Finally, numerical simulations of two distinct random circuits show excellent agreement, thereby demonstrating universality.