Scrambling the spectral form factor: unitarity constraints and exact results
arXiv:1702.04350 · doi:10.1103/PhysRevD.95.126008
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 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 as well as higher-dimensional versions of AdS/CFT.
33 pages, 6 figures, published version
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