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arXiv:1705.07597 · Physical Review Letters

Finite-Size Scaling of Out-of-Time-Ordered Correlators at Late Times

Yichen Huang, Fernando G. S. L. Brandão, Yongliang Zhang

Published Jul 1, 2019• 71 citations
arXivHTMLPDF

Abstract

Chaotic dynamics in quantum many-body systems scrambles local information so that at late times it can no longer be accessed locally. This is reflected quantitatively in the out-of-time-ordered correlator of local operators, which is expected to decay to 0 with time. However, for systems of finite size, out-of-time-ordered correlators do not decay exactly to 0 and in this paper we show that the residual value can provide useful insights into the chaotic dynamics. When energy is conserved, the late-time saturation value of the out-of-time-ordered correlator of generic traceless local operators scales as an inverse polynomial in the system size. This is in contrast to the inverse exponential scaling expected for chaotic dynamics without energy conservation. We provide both analytical arguments and numerical simulations to support this conclusion.

DOI
10.1103/physrevlett.123.010601

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Chaotic dynamics in quantum many-body systems scrambles local information so that at late times it can no longer be accessed locally. This is reflected quantitatively in the out-of-time-ordered correlator of local operators, which is expected to decay to 0 with time. However, for systems of finite size, out-of-time-ordered correlators do not decay exactly to 0 and in this paper we show that the residual value can provide useful insights into the chaotic dynamics. When energy is conserved, the late-time saturation value of the out-of-time-ordered correlator of generic traceless local operators scales as an inverse polynomial in the system size. This is in contrast to the inverse exponential scaling expected for chaotic dynamics without energy conservation. We provide both analytical arguments and numerical simulations to support this conclusion.
The decay of out-of-time-ordered correlators (OTOCs) to a small residual value signals the scrambling of quantum information. This behavior is generic: it occurs in both chaotic and interacting integrable many-body systems, and therefore cannot by itself distinguish between them. In this paper we analytically study the late-time saturation value of a closely related observable, operator purity, and show that its scaling with system size robustly distinguishes chaotic from integrable systems. Our approach is based on the Mazur bound, which provides concrete scaling estimates and makes explicit why the locality of integrable charges matters. We support our findings with numerical simulations of one-dimensional spin chains.
A closed quantum system never forgets its initial state, but the encoded information can get scrambled and become inaccessible without measuring a large fraction of all the system degrees of freedom. This scrambling can be diagnosed by studying the spatial spreading of initially local operators under the Heisenberg time evolution, and the decay of the out-of-time-ordered correlators (OTOC). What insights can OTOCs provide to understand the dynamics of quantum many-body systems? What are the characteristic behaviors of OTOCs during the time evolution? How is information scrambling affected by the dissipation in open quantum many-body systems? We first study slow scrambling in many-body localized systems via calculating various correlators, two-point retarded correlators and OTOCs. Comparing with retarded correlators, OTOCs provide more information about the dynamics. We find that disorder slows and partially halts the onset of information scrambling. Instead of ballistic spreading, propagation of information forms a logarithmic light cone. Next, we study the finite-size scaling of OTOCs at late times in generic thermalizing quantum many-body systems. When energy is conserved, the late-time saturation value of the OTOC of generic traceless local operators scales as an inverse polynomial in the system size. This is in contrast to the inverse exponential scaling expected for chaotic dynamics without energy conservation. We also study information scrambling in open quantum many-body systems. We define a dissipative version of OTOC and study its behaviors in a prototypical chaotic quantum chain with dissipation. We find that dissipation leads to not only the overall decay of the scrambled information due to leaking, but also structural changes so that the information light cone can only reach a finite distance even when the effect of overall decay is removed. Finally, we construct a family of local Hamiltonians for understanding the asymmetric information scrambling. Our models live on a one-dimensional lattice and exhibit asymmetric butterfly light cone between the left and right spatial directions.
We provide bounds on temporal fluctuations around the infinite-time average of out-of-time-ordered and time-ordered correlators of many-body quantum systems without energy gap degeneracies. For physical initial states, our bounds predict the exponential decay of the temporal fluctuations as a function of the system size. We numerically verify this prediction for chaotic and interacting integrable spin-1/2 chains, which satisfy the assumption of our bounds. On the other hand, we show analytically and numerically that for the XX model, which is a noninteracting system with gap degeneracies, the temporal fluctuations decay polynomially with system size for operators that are local in the fermion representation and decrease exponentially in the system size for non-local operators. Our results demonstrate that the decay of the long-time temporal fluctuations of correlators cannot be used as a reliable metric of chaos or lack thereof.
Out-of-time-order correlators (OTOCs) have become established as a tool to characterise quantum information dynamics and thermalization in interacting quantum many-body systems. It was recently argued that the expected exponential growth of the OTOC is connected to the existence of correlations beyond those encoded in the standard Eigenstate Thermalization Hypothesis (ETH). We show explicitly, by an extensive numerical analysis of the statistics of operator matrix elements in conjunction with a detailed study of OTOC dynamics, that the OTOC is indeed a precise tool to explore the fine details of the ETH. In particular, while short-time dynamics is dominated by correlations, the long-time saturation behavior gives clear indications of an operator-dependent energy scale ω_{GOE} associated to the emergence of an effective Gaussian random matrix theory. We provide an estimation of the finite-size scaling of ω_{GOE} for the general class of observables composed of sums of local operators in the infinite-temperature regime and found linear behavior for the models considered.
We describe the dynamics of many-body quantum chaotic systems at all time scales by studying the Green's and out-of-time order correlation (OTOC) functions of the four-body, NNN-Majorana Sachdev-Ye-Kitaev model. By combining the scramblon formalism and random-matrix-theory techniques, we obtain analytical expressions for these functions at all times. The early exponential growth of the OTOC is followed by an exponential decay at a rate governed by that of the Green's function (the real part of the leading complex Ruelle-Pollicott resonances). For late times that scale exponentially with NNN, both functions have a dip-ramp-plateau pattern for Nmod8=2,6N \mathrm{mod}8 = 2, 6Nmod8=2,6 that deviates substantially from the ergodic prediction due to local-in-energy correlations of matrix elements and eigenvalues, even after the Heisenberg time.
We study out-of-time-order correlators (OTOCs) of the form $\ensuremath{\langle}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}(t)\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{B}(0)\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{C}(t)\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{D}(0)\ensuremath{\rangle}$ for a quantum system weakly coupled to a dissipative environment. Such an open system may serve as a model of, e.g., a small region in a disordered interacting medium coupled to the rest of this medium considered as an environment. We demonstrate that for a system with discrete energy levels the OTOC saturates exponentially $\ensuremath{\propto}\ensuremath{\sum}{a}_{i}{e}^{\ensuremath{-}t/{\ensuremath{\tau}}_{i}}+\mathrm{const}$ to a constant value at $t\ensuremath{\rightarrow}\ensuremath{\infty}$, in contrast with quantum-chaotic systems which exhibit exponential growth of OTOCs. Focusing on the case of a two-level system, we calculate microscopically the decay times ${\ensuremath{\tau}}_{i}$ and the value of the saturation constant. Because some OTOCs are immune to dephasing processes and some are not, such correlators may decay on two sets of parametrically different time scales related to inelastic transitions between the system levels and to pure dephasing processes, respectively. In the case of a classical environment, the evolution of the OTOC can be mapped onto the evolution of the density matrix of two systems coupled to the same dissipative environment.