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

TL;DR

This work investigates chaos in the 1D Bose-Hubbard system via out-of-time-order correlators (OTOCs) to probe quantum critical dynamics. Using exact diagonalization, the authors show the Lyapunov exponent $\lambda_L$ exhibits a broad peak near the quantum critical region at finite temperature, approaching the chaos bound $\lambda_L \le 2\pi/\beta$, consistent with a proposed Quantum Critical Point (QCP) Conjecture that critical fluctuations maximize scrambling. They connect OTOC growth to second Rényi entropy growth through the OTOC-RE theorem and provide a conformal-field-theory expression $S_A^{(2)}(t)=\frac{c}{8}\log(\sinh(\pi T t))$ with asymptotic linear-in-time behavior, supporting exponential OTOC deviations. The paper also outlines a practical two-copy interference protocol to measure OTOCs without Hamiltonian inversion and discusses the butterfly velocity $v_B$ and finite-size effects, highlighting experimental routes in cold-atom Bose-Hubbard setups to test quantum-chaotic dynamics near criticality.

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 $2π/β$ 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.

Figures (6)

• Figure 1: (a) Schematic phase diagram of the Bose-Hubbard model. The dotted line illustrates the parameter regime that is considered in this work. (b) Schematic OTOC and the fitting scheme to obtain the Lyapunov exponent. See Sec. \ref{['lya']} for more details.
• Figure 2: The growth of the second Rényi entropy $S_A^{(2)}$ and the normalized OTOC $|\tilde{F}(t)|$ as functions of time $tJ$ for $U/J=10$ at $\beta J=0.9$ and $N=L=6$ with a periodic boundary condition. The linear growth regime of $S_A^{(2)}$ is indicated by a fitted dashed black line. See the main text for more details on the operator choice.
• Figure 3: (a) The amplitude of normalized OTOC $|\tilde{F}(t)|$ as a function of time $tJ$ for $U/J=4, 6$ and $8$ at $\beta J=0.9$ and $N=L=7$. $N$ is the number of bosons and $L$ is the system size. The inset is a zoom-in plot of the early-time deviation behavior with $t_0$ aligned together. It is clear that the $U/J=6$ curve deviates faster than the $U/J=4$ and $8$ curves. (b-c) The Lyapunov exponents as a function of $U/J$. The error bars come from the fitting. (b) is plotted for $\beta J=0.9$ and $0.2$ with $N=L=7$; (c) is plotted for $N=7$ and $N=3$ with $L=7$, $\beta J=0.9$. In all the three figures above, we have chosen $\hat{V}=\hat{b}_1$, $\hat{W}=\hat{b}_4$ and the periodic boundary condition. For the fitting, we take the fitting parameters $F_\mathrm{c}=0.99$ and $p=0.2$. We have verified that changing the fitting parameters will not affect the trend of the data, but will only modify the exponents quantitatively.
• Figure 4: (a) The amplitude of normalized OTOC $|\tilde{F}(t)|$ as a function of time $tJ$ for $U/J=1,3$ and $5$ at $\beta J=1.0$. (b) The Lyapunov exponents as a function of $U/J$ plotted for $\beta J=1.0$ and $0.4$. The error bars come from the fitting. In all the two figures above, we have chosen $N=L=6$ with a periodic boundary condition, $\hat{V}=\hat{W}=\hat{b}_{\bf k}$, $k=\pi/3$. For the fitting, we take the fitting parameters $F_\mathrm{c}=0.99$ and $p=0.8$. We have verified that changing the fitting parameters will not affect the trend of the data, but will only modify the exponents quantitatively.
• Figure 5: (a) The amplitude of normalized OTOC $|\tilde{F}(t)|$ as a function of time $tJ$ for $U/J=6$. $\hat{V}=\hat{b}_i$ and $\hat{W}=\hat{b}_j$ with $i$ fixed at $i=1$ and $j$ varies between $j=2$, $j=3$ and $j=4$. (b) The butterfly velocity extracted from the OTOC. $a_0$ is the lattice spacing. The inset is the time $t_0$ where the OTOC begins to deviate exponentially as a function of the site $j$ for $U/J=6$. In all the two plots above, $\beta J=0.9$ and $N=L=7$ with periodic boundary condition. To extract $t_0$ we choose $F_{\mathrm{c}} =0.99$.