Statistics of the Random Matrix Spectral Form Factor

TL;DR

We investigate the spectral form factor (SFF) in random matrix theory for the circular (CUE) and Gaussian (GUE) ensembles in the ergodic regime, focusing on subleading corrections in the matrix dimension $D$. The study combines two complementary methods — sine-kernel random-matrix techniques and a four-fold replicated supersymmetric sigma-model — to derive the next-to-leading order statistics of the SFF, confirming a universal non-Gaussian correction ${Conn}_2(t) = z(2t) - 2 z(t)$ and clarifying ensemble-specific refinements. The leading behavior remains Gaussian with moments $ig rbracket { m SFF}^n ig rbracket o n! [D z(t)]^n$, while non-perturbative saddle points in the SUSY framework reveal the precise form of subleading corrections and explain time-domain kinks near $ au = t/t_H = 1/2$ and $1$. Numerical simulations across $D$ and both ensembles corroborate the theory, providing a robust null-model for SFF fluctuations in chaotic quantum systems and guiding future extensions to higher-order corrections and other symmetry classes.

Abstract

The spectral form factor of random matrix theory plays a key role in the description of disordered and chaotic quantum systems. While its moments are known to be approximately Gaussian, corrections subleading in the matrix dimension, $D$, have recently come to attention, with conflicting results in the literature. In this work, we investigate these departures from Gaussianity for both circular and Gaussian ensembles. Using two independent approaches -- sine-kernel techniques and supersymmetric field theory -- we identify the form factor statistics to next leading order in a $D^{-1}$ expansion. Our sine-kernel analysis highlights inconsistencies with previous studies, while the supersymmetric approach backs these findings and suggests an understanding of the statistics from a complementary perspective. Our findings fully agree with numerics. They are presented in a pedagogical way, highlighting new pathways (and pitfalls) in the study of statistical signatures at next leading order, which are increasingly becoming important in applications.

Figures (8)

• Figure 1: Numerical summary of the paper. The spectral form factor (SFF) of a single realization (red dots) fluctuates over time around its average $\overline{\rm SFF}$ (black line), with a variance ${\rm Var(SFF)}=\overline{\rm SFF^2}-\overline{\rm SFF}^2$ proportional to the average in the large-$D$ limit (blue line), i.e. Gaussian distribution. The amplitude of these fluctuations is indicated by the shaded red region. The data show deviations from the Gaussian behavior for $\tau = t/t_{\rm H} > 1/2$, as emphasized in the inset, which shows $D^{-1}|\overline{{\mathrm{SFF}}^2}-2\overline{{\mathrm{SFF}}}^2|$. This work addresses precisely such deviations from Gaussianity, providing two analytical frameworks to account for them. Numerical simulation from a CUE for $D=20$, average with $N_{\rm sim}=2 \cdot 10^7$ samples.
• Figure 2: Leading (a-b) and subleading (c-d) behavior of the second moment of the spectral form factor. Results were sampled from the CUE (left) and for the evolution of a GUE Hamiltonian (right). The numerical data for $D=5, 10, 100$ (red lines with increasing color intensity) are compared with the analytical prediction [$2 z(\tau)^2$ (a-b) and $2z(\tau)-z(2\tau)$ (c-d)] (dashed black) using Eq. \ref{['eq_haake']} and Eq. \ref{['eq_brezin']} in the two panels. The empirical average is obtained with $N_{\rm sim}=(5 \cdot 10^4, 5 \cdot 10^5, 2 \cdot 10^6)$ with $D=5, 10, 100$ respectively.
• Figure 3: Higher-order moments of the spectral form factor, $\overline{{\mathrm{SFF}}^n}$ for $n=1,2,3,4$ (orange lines with increasing intensity) compared with the Gaussian prediction, cf. Eq. \ref{['eq_gauss_moment']} (black dashed line). Results are sampled from the CUE (left panel) and for the evolution of a GUE Hamiltonian (right panel) using $z(t)$ from Eq. \ref{['eq_haake']} and Eq. \ref{['eq_brezin']} in the two panels. The numerical data at fixed $D=10$ averaged over $N_{\rm sim} = 5\cdot 10^5$ samples.
• Figure 4: Oscillations in the subleading correction to the second moment of the spectral form factor for different empirical averages over $N_{\rm sim}$ simulations. Results are sampled from the CUE (left panel) and for the evolution of a GUE Hamiltonian (right panel). The numerical data at fixed $D=100$ averaged over $N_{\rm sim}=20, 2\cdot 10^4, 2\cdot 10^6$ (blue lines with increasing color intensity) are compared with the prediction in Eq. \ref{['eq_main']}, with full blue lines we indicate the amplitude of the oscillatory term $\mathcal{O}(D z^2(\tau)/\sqrt{N_{\rm sim}})$ and with a dashed black line the subleading correction in Eq. \ref{['eq_conn']}, using Eq. \ref{['eq_haake']} and Eq. \ref{['eq_brezin']} in the two panels.
• Figure 5: Top left: graphical representation of the expansion of the resolvent in Eq. \ref{['eq_resolvent']} to first order in $H$. Top right: The averaging of a second order term leads to index correlations as indicated. Right center: To leading order in the $D^{-1}$ expansion, each additional order in $H^{2}$ must be accompanied by a free running index summation. The diagram shown satisfies this criterion. Right bottom: a suppressed diagram, which we disregard. Bottom: the recursive summation of all diagrams without crossing lines defines the SCBA.