Out-of-Time-Order Correlation for Many-Body Localization

TL;DR

This paper investigates how out-of-time-order correlators (OTOC) reveal non-ergodic dynamics in many-body localized (MBL) systems and distinguish them from Anderson localization (AL). It analyzes both a phenomenological l-bit model and a random-field XXZ chain, finding that the OTOC decays with a power law in the MBL phase and remains constant in AL, while entanglement growth in MBL is logarithmic and correlated with OTOC decay. The authors prove a general OTOC–Rényi entropy (RE) theorem connecting the growth of the second Rényi entropy after a quench to a sum of equilibrium OTOCs, valid for generic quantum systems and extendable to finite temperature. They verify the theorem with explicit calculations: for the phenomenological model, exp(−S_A^{(2)}) matches the summed OTOCs, and a similar agreement holds in the random-field XXZ model. The results provide a robust, experimentally accessible diagnostic of MBL and link non-equilibrium entropy production to equilibrium correlation functions, with potential testing in cold-atom and trapped-ion experiments.

Abstract

In this paper we first compute the out-of-time-order correlators (OTOC) for both a phenomenological model and a random-field XXZ model in the many-body localized phase. We show that the OTOC decreases in power law in a many-body localized system at the scrambling time. We also find that the OTOC can also be used to distinguish a many-body localized phase from an Anderson localized phase, while a normal correlator cannot. Furthermore, we prove an exact theorem that relates the growth of the second Rényi entropy in the quench dynamics to the decay of the OTOC in equilibrium. This theorem works for a generic quantum system. We discuss various implications of this theorem.

Figures (3)

• Figure 1: The calculation of the von Neumann entropy, the second Rényi entropy and the OTOC for the MBL and the AL cases in random-field XXZ model Eq. \ref{['XXZ']}. The OTOC has been rescaled to drop from unity. The horizontal axis is $tJ_\perp$ in the logarithmic scale. The calculation is done for on an $8$-site model with open boundary condition, and is averaged over $10^3$ disorder configurations. Here $J_\perp>0$, $h_i/J_\perp$ is uniformly distributed between $[-5,5]$. For the MBL case $J_z/J_\perp=0.2$ where the system is known to be fully localized mbl-transition. For the AL case $J_z=0$.
• Figure 2: Diagrammatic illustration of how to prove the OTOC-RE theorem. Please see the main text for details.
• Figure 3: (a) All the four OTOCs with $\hat{V}=(1+2\hat{s}^x_i)/2^D$ ($i=1$) and $W=1/\sqrt{2}$, $\sqrt{2}\hat{s}^x_j$, $\sqrt{2}\hat{s}^y_j$ and $\sqrt{2}\hat{s}^z_j$($j=8$), respectively. And all the OTOCs have been rescaled to drop from unity. (b) The summation of four (unrescaled) OTOCs, the second order Rényi entropy $S^{(2)}_\text{A}$ (trace out $j=8$ site which is the subsystem $B$) after quench by operator $\hat{O}=(1+2\hat{s}^x_i)/2^{(D+1)/2}$ and $\exp(-S^{(2)}_\text{A})$. Here the calculation is taken on an $8$-site chain, all quantities have been averaged over $10^3$ disorder configurations. $J_z/J_\perp=0.2$ and $h_i/J_\perp$ is uniformly distributed between $[-5,5]$.