2018/03/22 by Sagar Vijay, Ashvin Vishwanath, Vijay, Sagar +1 · 1 citation
Mathematics · Physics and Astronomy · #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Quantum Physics (quant-ph) #Random Matrices and Applications #Statistical Mechanics (cond-mat.stat-mech) #Stochastic processes and statistical mechanics #Strongly Correlated Electrons (cond-mat.str-el) #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1803.08483
openalex publication_date 2018/03/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the finite-temperature scrambling behavior of a quantum system described by a Hamiltonian chosen from a random matrix ensemble. This effectively (0+1)-dimensional model admits an exact calculation of various ensemble-averaged out-of-time-ordered correlation functions in the large-N limit, where N is the Hilbert space dimension. For a Hamiltonian drawn from the Gaussian unitary ensemble, we calculate the ensemble averaged OTOC at all temperatures. In addition to an early time quadratic growth of the averaged out-of-time-ordered commutator (OTOC), we determine that the OTOC saturates to its asymptotic value as a power-law in time, with an exponent that depends both on temperature, and on one of four classes of operators appearing in the correlation function, that naturally emerge from this calculation. Out-of-time-ordered correlation functions of operators that are distributed around the thermal circle take a time ts∼ β to decay at low temperatures. We generalize these exact results, by demonstrating that out-of-time-ordered correlation functions averaged over any ensemble of Hamiltonians that are invariant under unitary conjugation H → U H U†, exhibit power-law decay to an asymptotic value. We argue that this late-time behavior is a generic feature of unitary dynamics with energy conservation. Finally, by comparing the OTOC with a commutator-anticommutator correlation function, we examine whether there is a time window over which a typical Hamiltonian behaves as a "coherent scrambler" in the language of Ref. \citeKitaevIAS, KitaevSuh.