Higher-order spectral form factors of circular unitary ensemble

TL;DR

This paper derives exact formulas for second- and third-order spectral form factors in the circular unitary ensemble, revealing their asymptotic behaviors and growth patterns across different regimes, advancing understanding of quantum chaos and universality.

Contribution

It provides the first exact closed-form expressions for higher-order spectral form factors in CUE for all time values, including asymptotic analysis.

Findings

Second-order SFF grows logarithmically with ensemble size at certain times

Exact formulas involve polygamma functions for second-order SFF

Third-order SFF derived in a special case using translational invariance

Abstract

Spectral form factor (SFF), one of the key quantity from random matrix theory, serves as an important tool to probe universality in disordered quantum systems and quantum chaos. In this work, we present exact closed-form expressions for the second- and third-order SFFs in the circular unitary ensemble (CUE), valid for all real values of the time parameter, and analyze their asymptotic behavior in different regimes. In particular, for the second-order SFF, we derive an exact closed-form expression in terms of polygamma functions. In the limit of infinite matrix size, and when the time parameter is restricted to integer values, the second-order SFF reproduces the standard result established in earlier studies. When the time parameter is of order one relative to the matrix size, we demonstrate that the second-order SFF grows logarithmically with the ensemble dimension. For the third-order SFFs, a closed-form result in a special case is obtained by exploiting the translational invariance of CUE.