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arXiv:1807.06086 · Physical Review Letters

Scrambling Dynamics across a Thermalization-Localization Quantum Phase Transition

Subhayan Sahu, Shenglong Xu, Brian G. Swingle

Published Oct 18, 2019• 54 citations
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Abstract

We study quantum information scrambling, specifically the growth of Heisenberg operators, in large disordered spin chains using matrix product operator dynamics to scan across the thermalization-localization quantum phase transition. We observe ballistic operator growth for weak disorder, and a sharp transition to a phase with subballistic operator spreading. The critical disorder strength for the ballistic to subballistic transition is well below the many body localization phase transition, as determined from finite size scaling of energy eigenstate entanglement entropy in small chains. In contrast, we find that the transition from subballistic to logarithmic behavior at the actual eigenstate localization transition is not resolved in our finite numerics. These data are discussed in the context of a universal form for the growing operator shape and substantiated with a simple phenomenological model of rare regions.

DOI
10.1103/physrevlett.123.165902

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We study quantum information scrambling, specifically the growth of Heisenberg operators, in large disordered spin chains using matrix product operator dynamics to scan across the thermalization-localization quantum phase transition. We observe ballistic operator growth for weak disorder, and a sharp transition to a phase with subballistic operator spreading. The critical disorder strength for the ballistic to subballistic transition is well below the many body localization phase transition, as determined from finite size scaling of energy eigenstate entanglement entropy in small chains. In contrast, we find that the transition from subballistic to logarithmic behavior at the actual eigenstate localization transition is not resolved in our finite numerics. These data are discussed in the context of a universal form for the growing operator shape and substantiated with a simple phenomenological model of rare regions.
Understanding the spread of quantum entanglement and scrambling of information across a quantum many-body system is a fundamental problem in quantum dynamics. A necessary part of the spread of the entanglement is the growth in time of Heisenberg operators of initially localized operators. This operator growth can be diagnosed by studying the space-time structure of commutators of well-separated operators. Previous studies in this area focused on translation invariant models, but the effects of disorder, which can radically alter the motion of heat and charge, had not been studied. Here, the authors study the growth of operators in interacting systems with quenched disorder. In the localized phase, they find that operator sizes grow logarithmically in time similar to other measures of entanglement spreading. In the disordered metal, they find that operator sizes grow linearly with time, i.e., ballistically, in contrast to the diffusive motion of charge and heat. The ballistic growth of operators is quantified by the butterfly velocity which the authors relate to the charge diffusion constant and the interaction-induced inelastic scattering rate. When the diffusion of charge is slow, the resulting butterfly velocity is much smaller than the maximum speed allowed by microscopic causality constraints.
We study the quantum thermalization and information scrambling dynamics of an experimentally realizable quantum spin model with homogeneous XX-type all-to-all interactions and random local potentials. We identify the thermalization-localization transition by changing the disorder strength, under a proper all-to-all interaction strength. The scrambling dynamics in the localization phase shows behaviors that are distinct from that of local models. The operator scrambling grows almost equally fast in the two phases. In the thermal phase, we show that fast scrambling is exhibited without appealing to the semiclassical limit. We also briefly discuss the experimental realization of the model using superconducting qubit quantum simulators.
Many-body localization is characterized by a slow logarithmic growth of the entanglement entropy after a global quantum quench while the local memory of an initial density imbalance remains at infinite time. We investigate how much the proximity of a many-body localized phase can influence the dynamics in the delocalized ergodic regime where thermalization is expected. Using an exact Krylov space technique, the out-of-equilibrium dynamics of the random-field Heisenberg chain is studied up to L=28L=28L=28 sites, starting from an initially unentangled high-energy product state. Within most of the delocalized phase, we find a sub-ballistic entanglement growth $S(t)\ensuremath{\propto}{t}^{1/z}$ with a disorder-dependent exponent $z\ensuremath{\ge}1$, in contrast with the pure ballistic growth z=1z=1z=1 of clean systems. At the same time, anomalous relaxation is also observed for the spin imbalance $\mathcal{I}(t)\ensuremath{\propto}{t}^{\ensuremath{-}\ensuremath{\zeta}}$ with a continuously varying disorder-dependent exponent $\ensuremath{\zeta}$, vanishing at the transition. This provides a clear experimental signature for detecting this nonconventional regime.
We investigate the phase transition between an ergodic and a many-body localized phase in infinite anisotropic spin-1/21/21/2 Heisenberg chains with binary disorder. Starting from the N\'eel state, we analyze the decay of antiferromagnetic order ms(t){m}_{s}(t)ms​(t) and the growth of entanglement entropy Sent(t){S}_{\mathrm{ent}}(t)Sent​(t) during unitary time evolution. Near the phase transition we find that ms(t){m}_{s}(t)ms​(t) decays exponentially to its asymptotic value ${m}_{s}(\ensuremath{\infty})\ensuremath{\ne}0$ in the localized phase while the data are consistent with a power-law decay at long times in the ergodic phase. In the localized phase, ${m}_{s}(\ensuremath{\infty})$ shows an exponential sensitivity on disorder with a critical exponent $\ensuremath{\nu}\ensuremath{\sim}0.9$. The entanglement entropy in the ergodic phase grows subballistically, ${S}_{\mathrm{ent}}(t)\ensuremath{\sim}{t}^{\ensuremath{\alpha}}$, $\ensuremath{\alpha}\ensuremath{\le}1$, with $\ensuremath{\alpha}$ varying continuously as a function of disorder. Exact diagonalizations for small systems, on the other hand, do not show a clear scaling with system size and attempts to determine the phase boundary from these data seem to overestimate the extent of the ergodic phase.
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The transition between many-body localized states and the delocalized thermal states is an eigen-state phase transition at finite energy density outside the scope of conventional quantum statistical mechanics. In this work we investigate the properties of the transition by studying the behavior of the entanglement entropy of a subsystem of size LAL_ALA​ in a system of size L>LAL > L_AL>LA​ near the critical regime of the many-body localization transition. The many-body eigenstates are obtained by exact diagonalization of a disordered quantum spin chain under twisted boundary conditions to reduce the finite-size effect. We present a scaling theory based on the assumption that the transition is continuous and use the subsystem size LA/ξL_A /ξLA​/ξ as the scaling variable, where ξξξ is the correlation length. We show that this scaling theory provides an effective description of the critical behavior and that the entanglement entropy follows the thermal volume law at the transition point. We extract the critical exponent governing the divergence of ξξξ upon approaching the transition point. We also study the participation entropy in the spin-basis of the domain wall excitations and show that the transition point and the critical exponent agree with those obtained from finite size scaling of the entanglement entropy. Our findings suggest that the many-body localization transition in this model is continuous and describable as a localization transition in the many-body configuration space.