Accessing scrambling using matrix product operators

Abstract

Scrambling, a process in which quantum information spreads over a complex quantum system becoming inaccessible to simple probes, happens in generic chaotic quantum many-body systems, ranging from spin chains, to metals, even to black holes. Scrambling can be measured using out-of-time-ordered correlators (OTOCs), which are closely tied to the growth of Heisenberg operators. In this work, we present a general method to calculate OTOCs of local operators in local one-dimensional systems based on approximating Heisenberg operators as matrix-product operators (MPOs). Contrary to the common belief that such tensor network methods work only at early times, we show that the entire early growth region of the OTOC can be captured using an MPO approximation with modest bond dimension. We analytically establish the goodness of the approximation by showing that if an appropriate OTOC is close to its initial value, then the associated Heisenberg operator has low entanglement across a given cut. We use the method to study scrambling in a chaotic spin chain with $201$ sites. Based on this data and OTOC results for black holes, local random circuit models, and non-interacting systems, we conjecture a universal form for the dynamics of the OTOC near the wavefront. We show that this form collapses the chaotic spin chain data over more than fifteen orders of magnitude.

Figures (9)

• Figure 1: (a) Light cone structure of a time-evolving ($U$) MPO, $U W U^\dagger$. The MPO starts as a local operator (red), and the initial required bond dimension is one. Under time evolution, higher bond dimension is only required in the shaded light cone region, since $U$ and $U^\dagger$ cancel outside the shaded region. Even inside the exact light cone, $U$ and $U^\dagger$ will approximately cancel for some time, so that the needed bond dimension remains small. (b) Entanglement of the MPO defined by treating it as a state with double the number of physical degrees. More generally, we can apply the MPO to a purified thermal state at temperature $T$ (the "thermofield double") and define a temperature dependent notion of operator entanglement. For systems with a holographic gravity dual (AdS/CFT), this temperature dependent operator entanglement is related to the entanglement between two halves of a two-sided wormhole. We show in Appendix \ref{['appsc:operator_entanglement']} that our results on operator entanglement are also valid in holographic systems at finite temperature.
• Figure 2: (a) The entanglement entropy growth of a typical Heisenberg operator, say $X(t)$, is restricted in a light cone. (b) For the mixed-field Ising model, $S$ grows sharply to its maximal value $\log(\chi)$ in a short time scale $\delta t$, while for the transverse-field Ising model, $S$ saturates to a much lower value bounded by $\log(4)$. We consider spin chains with $201$ spins and set the bond dimension $\chi=32$.
• Figure 3: (a) The early growth of the commutator of $X_r(t)$ and $X_{r'}$ with $r$=101 and $r'-r=0, 10, 20,\cdots,100$. An MPO with $\chi$ as small as $4$ is sufficient to capture the early growth region of $C$ from machine precision to $\sim 0.1$ across the whole chain. (b) The early growth behavior is fitted to $e^{-\lambda (x-v_Bt-x_0)^{1+p}/t^p}$ with $\lambda=3.8, p=0.67, v_B=0.67$ and $x_0=1.8$, indicating a sub-diffusively spreading wavefront.
• Figure 4: (a) The OTOC for the mixed-field model. The commutator exhibits two growth regions. In the second region, the growth is much slower and relaxes to the final value of $2$. The MPO result thus qualitatively matches expectations even inside the light cone. (b) In the case of the transverse-field model, an MPO with $\chi=4$ is sufficient to obtain $C(t)$ for arbitrarily long time because of the bounded entanglement. The commutator decays to $0$ in the late-time limit, showing no sign of scrambling as expected.
• Figure 5: Distinguishing the early-time behavior and early-growth behavior of the Bessel function. (a) In the early-time, the Bessel function $J_x(t)\sim \frac{(t/2)^x}{x!}$, which breaks down at the wavefront. (b) At the wavefront, the Bessel function collapses to the function with argument $(t-x)/t^{1/3}$. Bessel functions of order from 10 to 300 are shown.