Spectral Form Factor of Gapped Random Matrix Systems

TL;DR

The work analyzes the spectral form factor (SFF) in random-matrix-like systems with a macroscopic gap and a large degenerate ground-state sector. It shows that at low temperature the disconnected contribution dominates at all times, while the connected part depends only on the non-degenerate (random) sector; the Christoffel–Darboux kernel governs the analysis across Wishart, Bessel, and ${\rm N}=2$ JT/SJT models. A universal sine-kernel emerges in the truncated non-perturbative kernel as $\hbar\to 0$, setting the ramp slope and aligning with the leading double-trumpet results, and the gap-size sets the oscillation period in the disconnected part. The results yield concrete predictions for the ramp-to-plateau transition, including temperature and gap dependence, and point to broader implications for thermalization and black-hole microstate physics in gapped spectra.

Abstract

In this work, we study the spectral form factor of random matrix models which exhibit a large number of degenerate ground states accompanied by a macroscopic gap in the spectrum. The central aim of this work is to understand how the standard narrative about the behavior of the spectral form factor is modified in the presence of these parametrically large number of ground states. We show that, at sufficiently low temperatures, the spectral form factor is dominated by the disconnected contribution, even at arbitrarily late times. Moreover, we demonstrate that the connected form factor only depends on the eigenvalues of the non-degenerate sector. Using the Christoffel-Darboux kernel, we analyze a number of examples including the Bessel model and $\mathcal{N}=2$ Jackiw-Teitelboim supergravity. In these examples, we find damped oscillations in the disconnected form factor, with a period set by the inverse size of the gap. Furthermore, we demonstrate that the slope of the ramp in the connected form factor arises from a universal sine-kernel, which emerges from a truncation of the full non-perturbative kernel in the $\hbar \to 0$ limit, and find agreement with the leading double trumpet result. Finally, we present predictions for how the ramp will transition to a plateau in the connected form factor and demonstrate how the transition depends on the details of the leading spectral density of states.

Figures (12)

• Figure 1: We make plots of the normalized density of states as computed by $K_{sc.}(E,E)$ (blue line) for finite values of $N$ and compare to the Marchenko-Pastur PDF $\tilde{\rho}_{MP}(E)$ (yellow line) for $\tilde{\Gamma}=8$. We can see that as $N$ gets larger the $K_{sc.}(E,E)$ converges towards the Marchenko-Pastur PDF as claimed in Eq. (\ref{['MPDistribution']}). Note that the plots here only include the random sector the full density will include an extra delta function $\Gamma\delta(E)$ in the plot itself it will be a spike of height $\tilde{\Gamma}$ because we plot $\tilde{\rho}$ (everything divided by $N$).
• Figure 2: We plot the spectral form factor for the Wishart ensemble at $N=90$ at infinite (top and bottom left plots) and at finite temperature $T=E_{Gap}=E_-$ (top and bottom right plots) for various values of $\tilde{\Gamma}$. The orange line is the numeric computation of the ensemble averaged form factor generated by actually computing the averaged form factor from computer simulated eigenvalues over many trials. The dotted line is the computation of the disconnected part of the form factor which is obtained by numerically computing the integral given in Eq. (\ref{['DisconnFFKernFormula']}). The blue line is the spectral form factor associated with step approximation given by the expression in Eq. (\ref{['StepApproxSFF']}).
• Figure 3: The two left panels are plotting the time at which the $n-th$ peaks in the oscillations appear for the in the form factor appear at a temperature scale set by $E_{Gap}=E_-$ for different values of $\tilde{\Gamma}$. Next to them on the right side of the figure we compute the time difference between the $n$-th adjacent pair of peaks. We can see that the time difference between peaks approaches the value $\tau=\frac{2\pi}{E_{Gap}}=\frac{2\pi}{E_-}$ whose numerical value is represented by the red horizontal line.
• Figure 4: We compare computations of the form factor via the expression given by the analysis of the kernel (orange line) given in Eq. (\ref{['WishartConnFiniteNSFFAnalytic']}) to numeric computations given by computer generated eigenvalues averaged over many trials (blue line). We can see very close agreement. Deviations of the blue line from the yellow line are due to the finite number of trials used to generate the blue line.
• Figure 5: A 3D plot of $K_{sc.}(x_1,x_2)^2$ for $N=10,\tilde{\Gamma}=8$.