Scrambling the spectral form factor: unitarity constraints and exact results

TL;DR

This work formalizes the spectral form factor as the fidelity decay of a thermofield double, using quantum speed limits to bound early-time scrambling and Paley–Wiener/recurrence arguments for long-time behavior. It provides exact, controllable results across models: Gaussian unitary ensemble yields a dip–ramp–plateau structure with finite-N fidelity computable via orthogonal polynomials, while integrable cases (HO, xp-AdS2, Calogero–Sutherland) show periodic or zero-entropy-like features. In holographic contexts (AdS/CFT), the study connects these dynamics to black hole information loss and recovery, predicting Gaussian initial decay, power-law late-time tails, and RMT-like ramps at late times with central-charge scaling. The results offer a unified framework linking nonequilibrium quantum dynamics, spectral properties, and gravity duals, with implications for scrambling, information preservation, and quantum chaos.

Abstract

Quantum speed limits set an upper bound to the rate at which a quantum system can evolve and as such can be used to analyze the scrambling of information. To this end, we consider the survival probability of a thermofield double state under unitary time-evolution which is related to the analytic continuation of the partition function. We provide an exponential lower bound to the survival probability with a rate governed by the inverse of the energy fluctuations of the initial state. Further, we elucidate universal features of the non-exponential behavior at short and long times of evolution that follow from the analytic properties of the survival probability and its Fourier transform, both for systems with a continuous and a discrete energy spectrum. We find the spectral form factor in a number of illustrative models, notably we obtain the exact answer in the Gaussian unitary ensemble for any $N$ with excellent agreement with recent numerical studies. We also discuss the relationship of our findings to models of black hole information loss, such as the Sachdev-Ye-Kitaev model dual to AdS$_2$ as well as higher-dimensional versions of AdS/CFT.

Figures (2)

• Figure 1: Survival probability of the thermofield double state in the GUE. (a) The exact survival probability of the TDS is displayed as a function of time for $\lambda\beta=0.1,1,3$, from bottom to top. During its decay, $S(\beta,t)$ exhibits oscillations as a function of time, the amplitude of which diminishes with the temperature. In the low temperature regime the decay is approximately monotonic, and governed by a power law $t^{-3}$. (b) Energy fluctuations in a TDS as a function of the inverse temperature $\beta$. (c) Long-time quantum reconstruction of the TDS under unitary dynamics. The exact survival probability of the TDS is compared in logarithmic scale with the probability of the memory term. At long-times of evolution the memory term accurately reproduces the survival probability indicating that the exact evolution is consistent with the decay of the TDS in a classical probabilistic sense at intermediate times and its subsequent reconstruction. The dynamics is associated with a GUE random matrix ensemble at large $N$ with a density of states described by Wigner semicircle law with $\lambda\beta=0.1$. The memory term is evaluated at $\tau=t/2$.
• Figure 2: Exact survival probability in the GUE. a) Zero-temperature for GUE of size $N=105$: A period of decay and approximate revivals with a power-law envelope that scales as $1/t^3$ terminates at a dip, followed by a linear rise that saturates at a plateau of order $1/N$. b) Comparison of zero-temperature curves for $N=10$ (blue), $N=20$ (orange) and $N=30$ (green). c) Comparison of different curves with $N=35$ and temperature $\beta = 0,0.05,0.1,0.2,0.4$, from bottom to top. As the inverse temperature increases, the initial decay approaches a straight line, while the dip, ramp and plateau phase remain relatively stable.