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Out-of-Time-Ordered Correlator in Many-Body Localized Systems

Yichen Huang, Yongliang Zhang, Xie Chen

Published Aug 3, 2016• 10 citations
OpenAlex

Abstract

In many-body localized systems, propagation of information forms a light cone that grows logarithmically with time. However, local changes in energy or other conserved quantities are expected to spread only within a finite distance. Is it possible to detect the logarithmic light cone using local observables, starting from thermal states or eigenstates? Here we show that this is indeed the case, but only in an out-of-time-ordered way. In particular, we numerically calculate various correlators in a prototypical model of many-body localization. While correlators of local operators $A(t=0)$ and $B(t>0)$ ordered in time sequence -- e.g., $A(t=0)B(t>0)$ -- are not sensitive to the unbounded information propagation, those with out-of-time order -- e.g., $A(t=0)B(t>0)A(t=0)B(t>0)$ -- can detect the logarithmic light cone. We demonstrate the connection between out-of-time-ordered correlators and the Lieb-Robinson bound in many-body localized systems, and explain how to saturate the latter by measuring the involved commutator on specially designed initial states. Moreover, out-of-time-ordered correlators also allow us to study the temperature dependence of the light cones and we observe an increase of propagation velocity with increasing temperature.

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In many‐body localized systems, propagation of information forms a light cone that grows logarithmically with time. However, local changes in energy or other conserved quantities typically spread only within a finite distance. Is it possible to detect the logarithmic light cone generated by a local perturbation from the response of a local operator at a later time? We numerically calculate various correlators in the random‐field Heisenberg chain. While the equilibrium retarded correlator is not sensitive to the unbounded information propagation, the out‐of‐time‐ordered correlator can detect the logarithmic light cone. We relate out‐of‐time‐ordered correlators to the Lieb‐Robinson bound in many‐body localized systems, and show how to detect the logarithmic light cone with retarded correlators in specially designed states. Furthermore, we study the temperature dependence of the logarithmic light cone using out‐of‐time‐ordered correlators. image

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In many-body localized systems, propagation of information forms a light cone that grows logarithmically with time. However, local changes in energy or other conserved quantities are expected to spread only within a finite distance. Is it possible to detect the logarithmic light cone using local observables, starting from thermal states or eigenstates? Here we show that this is indeed the case, but only in an out-of-time-ordered way. In particular, we numerically calculate various correlators in a prototypical model of many-body localization. While correlators of local operators A(t=0)A(t=0)A(t=0) and B(t>0)B(t>0)B(t>0) ordered in time sequence -- e.g., A(t=0)B(t>0)A(t=0)B(t>0)A(t=0)B(t>0) -- are not sensitive to the unbounded information propagation, those with out-of-time order -- e.g., A(t=0)B(t>0)A(t=0)B(t>0)A(t=0)B(t>0)A(t=0)B(t>0)A(t=0)B(t>0)A(t=0)B(t>0) -- can detect the logarithmic light cone. We demonstrate the connection between out-of-time-ordered correlators and the Lieb-Robinson bound in many-body localized systems, and explain how to saturate the latter by measuring the involved commutator on specially designed initial states. Moreover, out-of-time-ordered correlators also allow us to study the temperature dependence of the light cones and we observe an increase of propagation velocity with increasing temperature.
In many‐body localized systems, propagation of information forms a light cone that grows logarithmically with time. However, local changes in energy or other conserved quantities typically spread only within a finite distance. Is it possible to detect the logarithmic light cone generated by a local perturbation from the response of a local operator at a later time? We numerically calculate various correlators in the random‐field Heisenberg chain. While the equilibrium retarded correlator is not sensitive to the unbounded information propagation, the out‐of‐time‐ordered correlator can detect the logarithmic light cone. We relate out‐of‐time‐ordered correlators to the Lieb‐Robinson bound in many‐body localized systems, and show how to detect the logarithmic light cone with retarded correlators in specially designed states. Furthermore, we study the temperature dependence of the logarithmic light cone using out‐of‐time‐ordered correlators. image
Lieb-Robinson bounds quantify the maximal speed of information spreading in nonrelativistic quantum systems.We discuss the relation of Lieb-Robinson bounds to out-of-time order correlators, which correspond to different norms of commutators C(r, t ) = [A i (t ), B i+r ] of local operators.Using an exact Krylov space-time evolution technique, we calculate these two different norms of such commutators for the spin-1/2 Heisenberg chain with interactions decaying as a power law 1/r α with distance r.Our numerical analysis shows that both norms (operator norm and normalized Frobenius norm) exhibit the same asymptotic behavior, namely, a linear growth in time at short times and a power-law decay in space at long distance, leading asymptotically to power-law light cones for α < 1 and to linear light cones for α > 1.The asymptotic form of the tails of C(r, t ) ∝ t/r α is described by short-time perturbation theory, which is valid at short times and long distances.
Study how quantum information propagates through spacetime manifold provides a means of identifying, distinguishing, and classifying novel phases of matter fertilized by many-body effects in strongly interacting systems in and out of equilibrium. Via a fuller characterization of key aspects regarding dynamic behaviors of information, we perform such an analysis on constrained many-body localization -- a newly proposed fully localized state under infinite-interaction limit -- in quasirandom Rydberg blockade spin chain models using thermal out-of-time-order commutators (OTOCs). The OTOC light cones predict a hitherto unknown Lieb-Robinson bound for constrained many-body localization, which is qualitatively different from that of unconstrained many-body Anderson insulators stabilized at weak-interaction limit. Our corroborated numeric and analytic study suggests that constrained many-body localization is a distinct dynamical eigenstate phase whose nonergodicity is beyond local-integral-of-motion phenomenology. Together, these findings consolidate the hierarchy of unconventional quantum dynamics encompassing constrained, unconstrained, and diagonal many-body-localized regimes.
Effective light cones, characterized by Lieb-Robinson bounds, emerge in nonrelativistic local quantum systems. Here, we present several analytical results derived from logarithmic light cones (LLCs). Possible origins of LLCs include the one-dimensional (1D) disordered XXZ model and a phenomenological model of many-body localization (MBL). In the LLC regime, we prove that, for arbitrary spatial dimensions and any initial pure state, entanglement growth is upper bounded by logarithmic time with an additional subleading double-logarithmic correction---arising from a real asymptotic solution of the Lambert W function---valid up to the asymptotic time limit. In the context of the 1D disordered XXZ model, this result resolves the ambiguity in distinguishing between logarithmic and power-law fits of entanglement growth in numerical studies; we also propose a falsifiable phenomenological functional form for the entanglement growth that agrees with existing numerical results. We show that information scrambling is logarithmically slow in the LLC regime. Furthermore, we demonstrate that the LLC supports long-lived quantum memories---quantum codes with macroscopic code distance and lifetimes that scale exponentially with system size---under unitary time evolution. Our analytical results provide benchmarks for future numerical studies of the MBL regime at large timescales.
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 investigate the operator entanglement of the time-evolution operator through the framework of eigenstate correlations. Focusing on strongly disordered quantum many-body systems in the many-body localized (MBL) regime, we analyze the operator entanglement across various spatiotemporal cuts, revealing the logarithmic lightcone of entanglement spreading. We demonstrate that this logarithmic lightcone arises directly from a hierarchy of energyscales and lengthscales encoded in eigenstate correlations. By characterizing the statistics of these hierarchical scales, we develop a microscopic theory for the spatiotemporal structure of entanglement spreading in MBL systems---without invoking phenomenological constructs such as $\ensuremath{\ell}$-bits. This approach reveals the fundamental connection between eigenstate correlations and the emergent entanglement structure in MBL systems.