  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.

Quantum Gases;Strongly Correlated Electrons