Topological field theory approach to intermediate statistics

TL;DR

This paper explores the spectral form factor of certain matrix ensembles linked to topological field theories, revealing intermediate statistics and transitions between ergodic and non-ergodic phases through topological invariants.

Contribution

It introduces a topological field theory approach to analyze intermediate statistics and ergodic transitions in matrix models related to Chern-Simons theory.

Findings

Spectral form factor proportional to HOMFLY invariant of torus links.

Ensembles exhibit intermediate statistics characteristic of ergodic-nonergodic transitions.

Topological tools effectively characterize phase transitions without local order parameters.

Abstract

Random matrix models provide a phenomenological description of a vast variety of physical phenomena. Prominent examples include the eigenvalue statistics of quantum (chaotic) systems, which are conveniently characterized using the spectral form factor (SFF). Here, we calculate the SFF of unitary matrix ensembles of infinite order with the weight function satisfying the assumptions of Szeg\"o's limit theorem. We then consider a parameter-dependent critical ensemble which has intermediate statistics characteristic of ergodic-to-nonergodic transitions such as the Anderson localization transition. This same ensemble is the matrix model of $U(N)$ Chern-Simons theory on $S^3$, and the SFF of this ensemble is proportional to the HOMFLY invariant of $(2n,2)$-torus links with one component in the fundamental and one in the antifundamental representation. This is one of a large class of ensembles arising from topological field and string theories which exhibit intermediate statistics. Indeed, the absence of a local order parameter suggests that it is natural to characterize ergodic-to-nonergodic transitions using topological tools, such as we have done here.