Universal Logarithmic Scrambling in Many Body Localization

Abstract

Out of time ordered correlator (OTOC) is recently introduced as a powerful diagnose for quantum chaos. To go beyond, here we present an analytical solution of OTOC for a non-chaotic many body localized (MBL) system, showing distinct feature from quantum chaos and Anderson localization (AL). The OTOC is found to fall only if the nearest distance between the two operators being shorter than $ξ\ln t$, where $ξ$ is dimensionless localization length. Thereafter, we found an universal power law decay of OTOC as $2^{-ξ\ln t}$, implying an universal logarithmic growth of second Rényi entropy, where $ξ$ plays the role of information scrambling rate. A relation between butterfly velocity and scrambling rate is found.

Figures (3)

• Figure 1: Illustration for quantum channel scheme for OTOC. A and D are output and input of the quantum circuit. The evolution time is $\tilde{t}$, shown in logarithmic scale. A logarithmic light-cone is drawn from inner side of A, where D enters the time-like region. Minimal distance between A and D is x.
• Figure 2: (a) one explicit calculation for $\hat{W}=[1]$ and $\hat{V}=[1]$ case. Red and blue dot are spin up and down. (b) Recursion strategy.
• Figure 3: In (a) we show OTOC $f(t)$ and second Rényi entropy $S_R^{(2)}$ for $(A|D)=(N|12\cdots N-1)$. Size of the system is taken to be $N=20$. The horizontal axis are all taken as $\xi\log t$. (b), we give OTOC for $(A|D)=(N|12\cdots M)$ with M=19,15,11. In (c) we the OTOC $f(t)$ and second Rényi entropy $S_R^{(2)}$ for $(A|D)=(N(N-1)\cdots\frac{N}{2}+1|12\cdots \frac{N}{2})$. In (d) we shows how Rényi entropy is different with different disorder distribution, the $\log t$ law is robust and the slopes are the same. (e) we give finite size scaling of the f(t) studied in (c). red for N=10, green for N=14 and blue for N=40.