Out-of-time-order correlation at a quantum phase transition

Motivated by the recent studies of out-of-time-order correlation functions and the holographic duality, we propose the quantum critical point conjecture, which is stated as: For a many-body quantum system with a quantum phase transition, the Lyapunov exponent extracted from the out-of-time-order correlators will exhibit a maximum around the quantum critical region. We first demonstrate that the Lyapunov exponent is well defined in the one-dimensional Bose-Hubbard model with the help of the out-of-time-order correlation–Rényi-entropy theorem. We then support the conjecture by numerically computing the out-of-time-order correlators. We also compute the butterfly velocity, and propose an experiment protocol of measuring this correlator without inverting the Hamiltonian.

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. 3 for more details.

Figure 2

The growth of the second Rényi entropy $S_A^{\left(2\right)}$ and the normalized OTOC $|\tilde{F}\left(t\right)|$ 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 $S_A^{\left(2\right)}$ 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 $|\tilde{F}\left(t\right)|$ 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 $\widehat{V}={\widehat{b}}_1$, $\widehat{W}={\widehat{b}}_4$ 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 $|\tilde{F}\left(t\right)|$ 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 the two figures above, we have chosen $N=L=6$ with a periodic boundary condition, $\widehat{V}=\widehat{W}={\widehat{b}}_{\mathbf{k}}$, $k=\pi /3$. 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 $|\tilde{F}\left(t\right)|$ as a function of time $tJ$ for $U/J=6$. $\widehat{V}={\widehat{b}}_i$ and $\widehat{W}={\widehat{b}}_j$ 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 $F_{\mathrm{c}}=0.99$.

Figure 6

The procedure of the conformal field theory calculation. (a) The original geometry on the stripe. There is a physical boundary along the imaginary axis and a cut along $+x$ direction from $z=i{\tau }_0$. For $\mathrm{tr}\left[{\rho }^n\right]$ we need $n$ copies and sew them together. (b) The complicated Riemann surface is identified with a twist field at $z=i{\tau }_0$ with $n$ copies of field on a single stripe. (c) After the conformal mapping, the problem becomes a standard geometry for a half infinite plane.