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

TL;DR

This study numerically investigates out-of-time-order correlations in the 1D Bose-Hubbard model, revealing a peak in the Lyapunov exponent near the quantum critical point, supporting the conjecture of maximal chaos at quantum phase transitions.

Contribution

It provides the first numerical evidence of Lyapunov exponent behavior across a quantum phase transition in the Bose-Hubbard model, linking chaos measures to quantum criticality.

Findings

Lyapunov exponent peaks at the quantum critical point

Out-of-time-order correlations exhibit exponential growth at scrambling time

Supports the conjecture of maximal chaos at quantum phase transitions

Abstract

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.