Out-of-time-ordered correlators in quantum Ising chain

Abstract

Out-of-time-ordered correlators (OTOC) have been proposed to characterize quantum chaos in generic systems. However, they can also show interesting behavior in integrable models, resembling the OTOC in chaotic systems in some aspects. Here we study the OTOC for different operators in the exactly-solvable one-dimensional quantum Ising spin chain. The OTOC for spin operators that are local in terms of the Jordan-Wigner fermions has a "shell-like" structure: after the wavefront passes, the OTOC approaches its original value in the long-time limit, showing no signature of scrambling; the approach is described by a $t^{-1}$ power law at long time $t$. On the other hand, the OTOC for spin operators that are nonlocal in the Jordan-Wigner fermions has a "ball-like" structure, with its value reaching zero in the long-time limit, looking like a signature of scrambling; the approach to zero, however, is described by a slow power law $t^{-1/4}$ for the Ising model at the critical coupling. These long-time power-law behaviors in the lattice model are not captured by conformal field theory calculations. The mixed OTOC with both local and nonlocal operators in the Jordan-Wigner fermions also has a "ball-like" structure, but the limiting values and the decay behavior appear to be nonuniversal. In all cases, we are not able to define a parametrically large window around the wavefront to extract the Lyapunov exponent.

Figures (12)

• Figure 1: (color online) The function $C_{xx}(\ell, t)$ for the quantum Ising chain at the critical point, $g = 1$, at infinite temperature (inverse temperature $\beta = 0$); the system size is $L = 512$. We show data as a function of $\ell$ at fixed time $t$, for $t$ in steps of $\Delta t = 2$ marked along the right border; here and in all figures, the energy unit $J$ in Eq. (\ref{['eqn:IsingModel']}) is set to $1$. The traces at fixed $t$ are shifted in the $y$ direction by $0.025 t$ thus offering three-dimensional-like visualization. For every $t$ that is a multiple of $10$, we mark the trace with red color for easier reading of the data. The light cone can be readily identified and corresponds to the maximal quasiparticle group velocity $c = \max_k \frac{d \epsilon_k}{d k} = J = 1$. In the timelike region, $C_{xx}(\ell, t)$ approaches zero in the long-time limit, indicating the absence of "scrambling."
• Figure 2: (color online) The function $C_{xx}(\ell, t)$ for several fixed separations $\ell$ at short time before the light cone reaches (i.e., spacelike separation between the operators). The growth of the commutator is compared to the "universal" power-law behavior given by $\approx 2 t^{2 (2\ell - 1)}/[(2\ell \!-\! 1)!]^2$; note that there is essentially no temperature dependence in this regime.
• Figure 3: The derivative function $G_{xx}(\ell, t) \equiv \partial \ln C_{xx}(\ell, t)/\partial t$ around the wavefront. Before the oscillation sets in, $G_{xx}(\ell, t)$ has very strong $\ell$ dependence, for which we do not know any universal description. Inset: $G_{xx}(\ell, t)$ for fixed $\ell = 40$ and several different inverse temperatures $\beta$, illustrating that there is basically no temperature dependence around the wavefront.
• Figure 4: Long-time behavior of $C_{xx}(\ell, t)$ in the timelike region; note the log-log scale. The data is shown as a function of $t$ at fixed $\ell$, where on the horizontal axis we show the time elapsed after the wavefront passes. Panel (a) shows several different separations $\ell$ and is at infinite temperature; the inset shows the same data on the linear-linear scale. Panel (b) shows several different temperatures at fixed separation $\ell = 20$. In all cases, we observe power-law decay $t^{-1}$, which can be understood from the long-time behavior of the fermion correlation functions.
• Figure 5: (color online) The function $C_{zz}(\ell, t) = 1 - \text{Re} F_{zz}(\ell, t)$ for the critical Ising chain ($g = 1$) at infinite temperature ($\beta = 0$), evaluated using the "doubling trick," Eq. (\ref{['doubleOTOC']}), on a periodic chain of length $L = 512$. Here we restore the sign of $\text{Re} F_{zz}(\ell, t)$ from $\text{Re} \sqrt{\Gamma_{zz}(\ell, t;L)}$ by requiring "continuity" of the "derivative" $D_\ell \text{Re} F_{zz}(\ell, t) = \text{Re} F_{zz}(\ell+1, t) - \text{Re} F_{zz}(\ell, t)$ (see text for details). We show data as a function of $\ell$ at fixed $t$, with time steps $\Delta t = 2$. The traces at fixed $t$ are shifted by $0.1 t$ in the $y$-direction for 3D-like visualization; every $t$ that is multiple of $10$ is marked with red color for easier tracing. Similarly to $C_{xx}(\ell, t)$ in Fig. \ref{['fig:XXotoc_all']}, we can readily identify the light cone and associate it with the maximal quasiparticle velocity $c = 1$. Unlike $C_{xx}(\ell, t)$, in the timelike region $C_{zz}(\ell, t)$ approaches a nonzero value close to 1 at long times. In other words, $F_{zz}(\ell, t)$ approaches value close to zero, which suggests scrambling of the information.