Decomposing the Spectral Form Factor

TL;DR

This work decomposes the spectral form factor (SFF) into contributions from k-th neighbor level spacings (k$n$LS), deriving analytical k-th neighbor SFFs (k$n$SFF) for GOE, GUE, GSE, and Poisson ensembles. It shows that k$n$SFFs consist of a Gaussian envelope with oscillatory components, with the envelope width governed by the spectral distance and Dyson index, and provides closed forms and asymptotics for both Gaussian and Poisson cases. The authors quantify minimal values, dip/Thouless times, and the deepest contributing neighbor k*, revealing distinct scaling: k* ~ N^{1/3} for Gaussian ensembles and k* ~ N^{1/2} for Poisson, and demonstrate how longer-range correlations shape the ramp and correlation hole. They validate the framework against random-matrix numerics and illustrate its relevance in the disordered XXZ chain, linking spectral statistics to dynamical signatures of chaos and integrability in many-body systems.

Abstract

Correlations between the energies of a system's spectrum are one of the defining features of quantum chaos. They can be probed using the Spectral Form Factor (SFF). We investigate how each spectral distance contributes in building this two-point correlation function. Specifically, starting from the spectral distribution of $k$-th neighbor level spacing ($k$nLS), we provide analytical expressions for the $k$-th neighbor Spectral Form Factor ($k$nSFF). We do so for the three Gaussian Random Matrix ensembles and the `Poissonian' ensemble of uncorrelated energy levels. We study the properties of the $k$nSFF, namely its minimum value and the time at which this minimum is reached, as well as the energy spacing with the deepest $k$nSFF. This allows us to quantify the contribution of each individual $k$nLS to the SFF ramp, which is a characteristic feature of quantum chaos. In particular, we show how the onset of the ramp, characterized either by the dip or the Thouless time, shifts to shorter times as contributions from longer-range spectral distance are included. Interestingly, the even and odd neighbors contribute quite distinctively, the first being the most important to built the ramp. They respectively yield a resonance or antiresonance in the ramp. All of our analytical results are tested against numerical realizations of random matrices. We complete our analysis and show how the introduced tools help characterize the spectral properties of a physical many-body system by looking at the interacting XXZ Heisenberg model with local on-site disorder that allows transitioning between the chaotic and integrable regimes.

Figures (16)

• Figure 1: Time evolution of the $k$nSFF for Poisson (black), GOE (red), GUE (green) and GSE (blue) for different spectral neighbors, $k = 1, 5, 20, 40$ in systems of dimension $N=100$. The plots for $k=5, \; 20, \; 40$ show analytical results (thin lines) from \ref{['SFFk_Pois']} and \ref{['SFFk_Approx']}, while for $k=1$ we show the exact expression \ref{['knSFF_Laguerre_final']}, and numerical results for random matrices averaged over $N_\mathrm{av}=1000$ realizations (thick transparent lines). While we do not expect the approximation \ref{['SFFk_Approx']} to be good for small $k$, it works already quite well for GSE and $k=1$ and less so for GUE and GOE, in that order. Note the different scales in the time axis, chosen to better represent the increasing number of oscillations with the spectral neighbor $k$, see Eq. \ref{['eq:nboscillations']}.
• Figure 2: Minimum time as a function of the neighbor degree for (a) ideal ensembles: Poisson (black), GOE (red), GUE (green) and GSE (blue) computed numerically from the unfolded spectrum along with the approximate expression \ref{['approx_td']} (dashed grey). (b) Results for the XXZ spin chain for different values of the disorder (blue colorscale) along with the analytical result $t_m(k) = \pi/k$ (dashed gray).
• Figure 3: (left) Minimum value of $k$nSFF $S_t^{(k)}$ as a function of the spectral distance $k$. Approximate analytical results (dashed lines) for the RMT ensembles \ref{['eq:minStd_RMT']} and Poisson distribution \ref{['eq:Std_Poi']}, and numerical results (solid lines). Colors are as in Fig. \ref{['fig:SFFk']}, i.e. Poisson (black), GOE (red), GUE (green) and GSE (blue). The numerical results are obtained from matrices of dimension $N=200$ and averaged over $N_\mathrm{av}=2000$ elements of the ensemble. Note that the function at large $k$ is linear, even if the choice of logarithmic scale in the $k$ axis does not allow a simple visualization. (right) Scaling of the deepest neighbor $k^*$ as a function of the system size $N$ computed numerically for RMT and Poisson ensembles (circles) and the analytical approximations \ref{['eq:kstar']} and \ref{['eq:kstar_poi']} rounded to the nearest integer (lines). Numerical results are averaged over $N_\mathrm{av}=200$ matrices. We show a guide for the eye at $N=200$ (gray dotted line) which agree with the values of $k^*$ used in Figure \ref{['fig:XXZ_Stdip']} for the Poissonian and GOE endpoints of $W$.
• Figure 4: Time scales for the partial $K$-neighbors SFF. The plots show the dip (pink), Thouless (turquoise), and plateau $t_p=2 \pi$ (black) times as a function of the maximum number of neighbors $K$ considered, for the three Gaussian ensembles. The shaded regions represent the part where the SFF grows in a non-universal way (pink) and where it grows with the universal ramp of the connected SFF (light blue). The results are computed with $N_\mathrm{en}=400$ from the analytical expressions of the SFF for Random Matrices \ref{['SFFk_Approx']}. For the Thouless time, we used the partial SFF without smoothing, taking $\epsilon = 0.1$ for GOE and GUE and $\epsilon = 0. 25$ for GSE. This choice is due to the challenge in building the full spike of the GSE from summing $k$nSFF's (cf. Apps. \ref{['app:dip_time_SFF']}, \ref{['App:SFF_test']}).
• Figure 5: Odd vs even neighbor contributions to the SFF and their sum for Poisson, GOE, GUE and GSE; computed numerically from $N_\mathrm{av}=1000$ matrices of dimension $N=100$. For visualization of the data in log-log scale, an extra factor of $1/2N$ was added to the even and odd contributions. The even contributions construct a 'resonance' while the odd ones construct an 'anti-resonance'. The vertical lines highlight the time at which the resonance and anti-resonance happen, $t^* = \pi$ (dashed gray), and at which the plateau starts for the Gaussian ensembles, $t_p = 2 \pi$ (dotted gray).