Out-of-Time-Order Correlation at a Quantum Phase Transition

In this paper we numerically calculate the out-of-time-order correlation functions in the one-dimensional Bose-Hubbard model. Our study is motivated by the conjecture that a system with Lyapunov exponent saturating the upper bound $2\pi/\beta$ will have a holographic dual to a black hole at finite temperature. We further conjecture that for a many-body quantum system with a quantum phase transition, the Lyapunov exponent will have a peak in the quantum critical region where there exists an emergent conformal symmetry and is absent of well-defined quasi-particles. With the help of a relation between the R\'enyi entropy and the out-of-time-order correlation function, we argue that the out-of-time-order correlation function of the Bose-Hubbard model will also exhibit an exponential behavior at the scrambling time. By fitting the numerical results with an exponential function, we extract the Lyapunov exponents in the one-dimensional Bose-Hubbard model across the quantum critical regime at finite temperature. Our results on the Bose-Hubbard model support the conjecture. We also compute the butterfly velocity and propose how the echo type measurement of this correlator in the cold atom realizations of the Bose-Hubbard model without inverting the Hamiltonian.