Spectral Form Factor of Gapped Random Matrix Systems

TL;DR

This paper investigates the spectral form factor in gapped random matrix models with many ground states, revealing how degeneracies and gaps alter late-time spectral behavior and wormhole contributions.

Contribution

It introduces a detailed analysis of spectral form factors in gapped systems, highlighting the dominance of disconnected contributions and the universal sine-kernel in the ramp.

Findings

Disconnected form factor dominates at low temperatures and late times.

Connected form factor depends only on non-degenerate eigenvalues.

Damped oscillations observed in the disconnected form factor.

Abstract

In this work, we study the spectral form factor of random matrix models which exhibit a large number of degenerate ground states accompanied by a macroscopic gap in the spectrum. The central aim of this work is to understand how the standard narrative about the behavior of the spectral form factor is modified in the presence of these parametrically large number of ground states. We show that, at sufficiently low temperatures, the spectral form factor is dominated by the disconnected contribution, even at arbitrarily late times. Moreover, we demonstrate that the connected form factor only depends on the eigenvalues of the non-degenerate sector, implying that BPS states do not contribute to wormhole calculations in the gravity context. Using the Christoffel-Darboux kernel, we analyze a number of examples including the Bessel model and $\mathcal{N}=2$ Jackiw-Teitelboim supergravity. In these examples, we find damped oscillations in the disconnected form factor, with a period set by the inverse size of the gap. Furthermore, we demonstrate that the slope of the ramp in the connected form factor arises from a universal sine-kernel, which emerges from a truncation of the full non-perturbative kernel in the $\hbar \to 0$ limit, and find agreement with the leading double trumpet result. Finally, we present predictions for how the ramp will transition to a plateau in the connected form factor and demonstrate how the transition depends on the details of the leading spectral density of states.