Reappearance of Thermalization Dynamics in the Late-Time Spectral Form Factor

TL;DR

This paper reveals a late-time deviation in the spectral form factor of quantum chaotic systems without time-reversal symmetry, linking it to early-time thermalization properties through the Riemann-Siegel lookalike formula.

Contribution

It introduces a second late-time deviation in the spectral form factor and derives a precise expression for it using the Riemann-Siegel lookalike formula, supported by numerical confirmation.

Findings

Late-time SFF deviates from RMT after Heisenberg time.

Deviation is determined by early-time thermalization.

Results confirmed numerically for various systems.

Abstract

The spectral form factor (SFF) is an important diagnostic of energy level repulsion in random matrix theory (RMT) and quantum chaos. The short-time behavior of the SFF as it approaches the RMT result acts as a diagnostic of the ergodicity of the system as it approaches the thermal state. In this work we observe that for systems without time-reversal symmetry, there is a second break from the RMT result at late time around the Heisenberg time. Long after thermalization has taken hold, and after the SFF has agreed with the RMT result to high precision for a time of order the Heisenberg time, the SFF of a large system will briefly deviate from the RMT behavior in a way exactly determined by its early time thermalization properties. The conceptual reason for this second deviation is the Riemann-Siegel lookalike formula, a resummed expression for the spectral determinant relating late time behavior to early time spectral statistics. We use the lookalike formula to derive a precise expression for the late time SFF for semi-classical quantum chaotic systems, and then confirm our results numerically for more general systems.