Sensitivity of the spectral form factor to short-range level statistics

TL;DR

The paper shows that self-correlation in short-range level statistics imposes exact constraints on the time-integrated spectral form factor $K(t)$, and that each expansion coefficient of the two-point function further fixes weighted time-integrals of $K(t)$. This complicates using $K(t)$ alone as an ergodicity probe, especially in systems with intermediate statistics; to address this, the authors propose a constraint-free diagnostic $\rho^{(2)}(0,0|t)=1-\frac{1}{\pi}\int_0^t(1-K(t'))\,dt'$, which tracks self-correlation without requiring comparison to fully ergodic baselines. They demonstrate the approach on randomly incomplete spectra and a Floquet model exhibiting many-body localization, showing that the new probe can reveal ergodicity properties where $K(t)$ alone is ambiguous. The work provides a more robust framework for diagnosing ergodicity and the onset of ergodicity across regimes, with potential implications for Thouless-energy scaling and MBL stability debates.

Abstract

The spectral form factor is a dynamical probe for level statistics of quantum systems. The early-time behaviour is commonly interpreted as a characterization of two-point correlations at large separation. We argue that this interpretation can be too restrictive by indicating that the self-correlation imposes a constraint on the spectral form factor integrated over time. More generally, we indicate that each expansion coefficient of the two-point correlation function imposes a constraint on the properly weighted time-integrated spectral form factor. We discuss how these constraints can affect the interpretation of the spectral form factor as a probe for ergodicity. We propose a new probe, which eliminates the effect of the constraint imposed by the self-correlation. The use of this probe is demonstrated for a model of randomly incomplete spectra and a Floquet model supporting many-body localization.

Figures (2)

• Figure 1: Numerically obtained $\rho^{(2)}(0,0|t)$ as a function of $t$ at system size $L=12$ for $\Gamma = 0.3$ (top curve), $\Gamma = 0.5$ (middle curve), and $\Gamma = 0.8$ (lower curve). The dashed line displays the random matrix (RMT) theory prediction for fully ergodic systems.
• Figure 2: Numerically obtained $\rho^{(2)}(0,0| 1000)$ as a function of $\Gamma$ for $L=8$ (squares), $L=10$ (circles), and $L=12$ (pentagons). For Poissonian and Wigner-Dyson level statistics, one expects respectively $\rho^{(2)}(0,0| 1000) = 1$ and $\rho^{(2)}(0,0| 1000) \approx 0.001$.