Spectral form factor in a random matrix theory

TL;DR

This paper investigates the spectral form factor in random matrix theory, focusing on oscillations near zero and the crossover point, using an exact contour-integral method to analyze finite-size effects and time dependence.

Contribution

It introduces an exact contour-integral approach to study the crossover oscillations of the spectral form factor in finite random matrices, extending analysis to time-dependent scenarios.

Findings

Oscillations near zero and crossover points are characterized.

Finite N effects on spectral correlations are analyzed.

Method extended to time-dependent spectral form factors.

Abstract

In the theory of disordered systems the spectral form factor $S(\tau)$, the Fourier transform of the two-level correlation function with respect to the difference of energies, is linear for $\tau<\tau_c$ and constant for $\tau>\tau_c$. Near zero and near $\tau_c$ its exhibits oscillations which have been discussed in several recent papers. In the problems of mesoscopic fluctuations and quantum chaos a comparison is often made with random matrix theory. It turns out that, even in the simplest Gaussian unitary ensemble, these oscilllations have not yet been studied there. For random matrices, the two-level correlation function $\rho(\lambda_1,\lambda_2)$ exhibits several well-known universal properties in the large N limit. Its Fourier transform is linear as a consequence of the short distance universality of $\rho(\lambda_1,\lambda_2)$. However the cross-over near zero and $\tau_c$ requires to study these correlations for finite N. For this purpose we use an exact contour-integral representation of the two-level correlation function which allows us to characterize these cross-over oscillatory properties. The method is also extended to the time-dependent case.