Quenched properties of the Spectral Form Factor

TL;DR

This work analyzes the quenched spectral form factor (SFF) for Hermitian and non-Hermitian random matrices, showing that while the SFF itself is not self-averaging, its quenched version is and agrees with the annealed counterpart up to subleading constants at high temperature. By recasting fluctuations in terms of a normalized complex partition function $\tilde{Z}$ and $Y=\log|\tilde{Z}|^2$, the authors derive a Gaussian-CLT description for $Z$ and a Gumbel distribution for $Y$ in the ramp/plateau regime, with deep downward spikes tied to zeros of $Z$ (Fisher zeros) producing exponential tails. The analysis is substantiated in several models, including the Random Energy Model (Poisson spectrum), a chaotic Hermitian spin glass, and a non-Hermitian generalization, and extended to non-Hermitian random matrices. The results recover and extend known late-time behavior and highlight the role of zeros in governing rare fluctuations, with potential implications for glassy phases and disordered quantum dynamics. The framework is shown to hold across Hermitian and non-Hermitian settings, suggesting broad applicability to quenched diagnostics in complex quantum systems.

Abstract

The Spectral Form Factor (SFF) is defined as the modulus squared of the partition function in complex temperature for hermitian matrices and a suitable generalisation has been given in the non hermitian case. In this work we compute the properties of the quenched SFF for hermitian and non hermitian random matrices. Despite the fact that the (annealed) SFF is not self-averaging the quenched SFF is self-averaging but these two averages coincide up to subleading constants (at least for high enough temperatures). The fluctuations of $\log \mathrm{SFF}$ are deep and one encounters thin spikes when moving close to a zero of the partition function. We study the partition function at late times by considering a suitable change of variable which turns out to be compatible with a Gumbel distribution. We note that the exponential tails of this distribution can be obtained by the deep spikes in the $\log \mathrm{SFF}$, namely the zeros of the partition function. We compare with the results obtained in isolated many-body systems and we show that same results hold at late times also for non-hermitian Hamiltonains and non-hermitian random matrices.

Figures (17)

• Figure 1: Top: Time sequence of $\mathrm{SFF}(t)/\langle \mathrm{SFF}(t)\rangle$ (upper panel) and $\ln \mathrm{SFF}(t)/\langle \ln \mathrm{SFF}(t)\rangle$ for the GOE for two sizes ${\mathcal{D}}=2^L$ and $L=6$ (red) and $L=13$ (blue) in the plateau regime. The deep spikes of $\ln \mathrm{SFF}(t)$ are induced by the proximity of a zero of $Z(\beta+i t)$, namely, anomalously small values of $\mathrm{SFF}$.
• Figure 2: Empirical Distribution of the variable $Y$ where the statistics are taken for a specific time over $10^5$ samples of spectra of the REM with $L=12$. The line in red corresponds to the Distribution of the variable $Y$ for $Z(\beta+it)\sim\mathcal{N}(0,\sigma^2)$.
• Figure 3: Time sequence of the quenched average (1000 samples) of the $\mathrm{SFF}(t)$ for the REM using various system sizes and $\beta=0$. Adding the quenched/annealed correction of Eq. (\ref{['eq:QA']}) in the inset, we see that the different curves match at the plateau given in Eq. (\ref{['SFF_quenched_REM']}).
• Figure 4: Empirical Distributions of the spectral gap ratios in the bulk of the spectrum of Eq. (\ref{['eq:XYSPINGLASS']}) for different system sizes and symmetry sectors using $10^5$ eigenvalues. In order of increasing system size, we use: $L= 8,10,12,$ and $14$.
• Figure 5: Time sequence of the quenched average of the $\mathrm{SFF}$ for different system sizes in Eq. (\ref{['eq:XYSPINGLASS']}). In order of increasing $L$, we use: $L= 8,10,12,$ and $14$. To have the curves match, we plot the normalized quenched average against $\ln{(t)}/\ln{(\mathcal{D})}$ and include the sub-dominant correction in Eq. (\ref{['eq:QA']}).