量子相变中的无序关联
Out-of-Time-Order Correlation at a Quantum Phase Transition
摘要：
本文数值计算了一维玻色-哈伯德模型中的无序关联函数。我们的研究是基于这样一个猜想：当李雅普诺夫指数达到上界$2π/β$时，系统在有限温度下与黑洞存在全息对偶。我们进一步推测，对于具有量子相变的多体量子系统，李雅普诺夫指数将在量子临界区出现一个峰值，在该临界区存在突现共形对称性，并且没有明确定义的准粒子.利用Rényi熵与无序关联函数之间的关系，证明了Bose-哈伯德模型的无序关联函数在置乱时刻也将呈现指数行为.通过对数值结果进行指数函数拟合，得到了有限温度下一维玻色-哈伯德模型在量子临界区的李雅普诺夫指数.我们在玻色-哈伯德模型上的结果支持了这个猜想。我们还计算了蝴蝶速度，并提出了如何在不反演哈密顿量的情况下，在玻色-哈伯德模型的冷原子实现中，这种相关器的回波型测量。
英文摘要：
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 $2π/β$ 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 Rényi 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.
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