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arXiv:1807.06039 · Physical Review Letters

Slow Growth of Out-of-Time-Order Correlators and Entanglement Entropy in Integrable Disordered Systems

Max McGinley, Andreas Nunnenkamp, Johannes Knolle

Published Jan 16, 2019• 46 citations
arXivHTMLPDF

Abstract

We investigate how information spreads in three paradigmatic one-dimensional models with spatial disorder. The models we consider are unitarily related to a system of free fermions and, thus, are manifestly integrable. We demonstrate that out-of-time-order correlators can spread slowly beyond the single-particle localization length, despite the absence of many-body interactions. This phenomenon is shown to be due to the nonlocal relationship between elementary excitations and the physical degrees of freedom. We argue that this nonlocality becomes relevant for time-dependent correlation functions. In addition, a slow logarithmic-in-time growth of the entanglement entropy is observed following a quench from an unentangled initial state. We attribute this growth to the presence of strong zero modes, which gives rise to an exponential hierarchy of time scales upon ensemble averaging. Our work on disordered integrable systems complements the rich phenomenology of information spreading and we discuss broader implications for general systems with nonlocal correlations.

DOI
10.1103/physrevlett.122.020603

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We investigate how information spreads in three paradigmatic one-dimensional models with spatial disorder. The models we consider are unitarily related to a system of free fermions and, thus, are manifestly integrable. We demonstrate that out-of-time-order correlators can spread slowly beyond the single-particle localization length, despite the absence of many-body interactions. This phenomenon is shown to be due to the nonlocal relationship between elementary excitations and the physical degrees of freedom. We argue that this nonlocality becomes relevant for time-dependent correlation functions. In addition, a slow logarithmic-in-time growth of the entanglement entropy is observed following a quench from an unentangled initial state. We attribute this growth to the presence of strong zero modes, which gives rise to an exponential hierarchy of time scales upon ensemble averaging. Our work on disordered integrable systems complements the rich phenomenology of information spreading and we discuss broader implications for general systems with nonlocal correlations.
When a group of compactly packed free fermions is allowed to spread over an empty one-dimensional lattice, the spreading particles can create entanglement between different parts of the lattice. We show, although breaking of translational invariance (TI) of the lattice by disorder slows down the spreading of local observables, the entanglement entropy of a subsystem can nonetheless receive a remarkable enhancement as long as the subsystem lies within the single-particle localization length. We show the main mechanism behind this enhancement is the reentrant exchange of particles between the subparts due to transport of mutual information due to backscattering. We discuss the length and timescales relevant to the phenomenon. We study the phenomenon for breaking of TI by both quasiperiodic and random potentials. We further explore the effect of randomness only in the initial state. This also exhibits a similar enhancement effect even in a TI lattice. We also touch upon the special case of periodic potential where qualitatively similar phenomenology emerges, although the coherence in the backscattering in this case leads to effects not captured by our simple yet generic picture.
The many-body localized phase of quantum systems is an unusual dynamical phase wherein the system fails to thermalize and yet, entanglement grows unboundedly albeit very slowly in time. We present a microscopic theory of this ultraslow growth of entanglement in terms of dynamical eigenstate correlations of strongly disordered, interacting quantum systems in the many-body localized regime. These correlations involve sets of four or more eigenstates and hence, go beyond correlations involving pairs of eigenstates which are usually studied in the context of eigenstate thermalization or lack thereof. We consider the minimal case, namely, the second R\'enyi entropy of entanglement, of an initial product state as well as that of the time-evolution operator, wherein the correlations involve quartets of four eigenstates. We identify that the dynamics of the entanglement entropy is dominated by the spectral correlations within certain special quartets of eigenstates. We uncover the spatial structure of these special quartets and the ensuing statistics of the spectral correlations amongst the eigenstates therein, which reveals a hierarchy of timescales or equivalently, energy scales. We show that the hierarchy of these timescales along with their nontrivial distributions conspire to produce the logarithmic in time growth of entanglement, characteristic of the many-body localized regime. The underlying spatial structures in the set of special quartets also provides a microscopic understanding of the spacetime picture of the entanglement growth. The theory therefore provides a much richer perspective on entanglement growth in strongly disordered systems compared to the commonly employed phenomenological approach based on the $\ensuremath{\ell}$-bit picture.
We study out-of-time ordered correlations C(x,t)C(x,t)C(x,t) and entanglement growth in the random-field XX model with open boundary conditions using the exact Jordan-Wigner transformation to a fermionic Hamiltonian. For any nonzero strength of the random field, this model describes an Anderson insulator. Two scenarios are considered: a global quench with the initial state corresponding to a product state of the N\'eel form, and the behavior in a typical thermal state at $\ensuremath{\beta}=1$. As a result of the presence of disorder, the information spreading as described by the out-of-time correlations stops beyond a typical length scale ${\ensuremath{\xi}}_{\text{OTOC}}$. For $|x|<{\ensuremath{\xi}}_{\text{OTOC}}$, information spreading occurs at the maximal velocity vmax=J{v}_{\mathrm{max}}=Jvmax​=J and we confirm predictions for the early-time behavior of $C(x,t)\ensuremath{\sim}{t}^{2|x|}$. For the case of the quench starting from the N\'eel product state, we also study the growth of the bipartite entanglement, focusing on the late- and infinite-time behavior. The approach to a bounded entanglement is observed to be slow for the disorder strengths we study.
Random fractal lattices provide a geometrically disordered setting in which quantum correlations can be shaped by noninteger dimensionality rather than onsite randomness. We investigate the entanglement properties of noninteracting fermions on random fractal lattices generated by a stochastic growth algorithm. By varying the growth parameter and adding missing links with probability ppp, we tune the Hausdorff and spectral dimensions while keeping the system free of onsite disorder. For ground states at different fillings, we compute the bipartite entanglement entropy of subregions defined by graph distance and analyze its scaling with subsystem size. Over a broad parameter range, we find robust power-law behavior governed primarily by the Hausdorff dimension, consistent with a generalized area law and without the logarithmic enhancement familiar from Euclidean free fermions. We also study entanglement growth following a global quench from an uncorrelated checkerboard state and uncover an asymptotic scaling collapse in which the subsystem-size dependence is governed by the Hausdorff dimension, while the temporal evolution is governed by the spectral dimension. The resulting dynamics are logarithmically slow over an extended intermediate-time window. These results show that geometric randomness alone can generate both nontrivial ground-state entanglement structure and slow quantum-information spreading in free-fermion systems.
Strong disorder often has drastic consequences for quantum dynamics. This is best illustrated by the phenomenon of Anderson localization in non-interacting systems, where destructive quantum wave interference leads to the complete absence of particle and information transport over macroscopic distances. In this work, we investigate the extent to which strong disorder leads to provably slow dynamics in many-body quantum lattice models. We show that in any spatial dimension, strong disorder leads to a non-perturbatively small velocity for ballistic information transport under unitary quantum dynamics, almost surely in the thermodynamic limit, in every many-body state. In these models, we also prove the existence of a "prethermal many-body localized regime", where entanglement spreads logarithmically slowly, up to non-perturbatively long time scales. More generally, these conclusions hold for all models corresponding to quantum perturbations to a classical Hamiltonian obeying a simple non-resonant condition. Deterministic non-resonant models are found, including spin systems in strong incommensurate lattice potentials. Consequently, quantum dynamics in non-resonant potentials is asymptotically easier to simulate on both classical or quantum computers, compared to a generic many-body system.
Recent numerical work by Bardarson, Pollmann, and Moore revealed a slow, logarithmic in time, growth of the entanglement entropy for initial product states in a putative many-body localized phase. We show that this surprising phenomenon results from the dephasing due to exponentially small interaction-induced corrections to the eigenenergies of different states. For weak interactions, we find that the entanglement entropy grows as ξln(Vt/ℏ), where V is the interaction strength, and ξ is the single-particle localization length. The saturated value of the entanglement entropy at long times is determined by the participation ratios of the initial state over the eigenstates of the subsystem. Our work shows that the logarithmic entanglement growth is a universal phenomenon characteristic of the many-body localized phase in any number of spatial dimensions, and reveals a broad hierarchy of dephasing time scales present in such a phase.