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arXiv:1902.00025 · Physical Review Letters

Localization as an Entanglement Phase Transition in Boundary-Driven Anderson Models

Michael J. Gullans, David A. Huse

Published Sep 10, 2019• 24 citations
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Abstract

The Anderson localization transition is one of the most well studied examples of a zero temperature quantum phase transition. On the other hand, many open questions remain about the phenomenology of disordered systems driven far out of equilibrium. Here we study the localization transition in the prototypical three-dimensional, noninteracting Anderson model when the system is driven at its boundaries to induce a current carrying nonequilibrium steady state. Recently we showed that the diffusive phase of this model exhibits extensive mutual information of its nonequilibrium steady-state density matrix. We show that this extensive scaling persists in the entanglement and at the localization critical point, before crossing over to a short-range (area-law) scaling in the localized phase. We introduce an entanglement witness for fermionic states that we name the mutual coherence, which, for fermionic Gaussian states, is also a lower bound on the mutual information. Through a combination of analytical arguments and numerics, we determine the finite-size scaling of the mutual coherence across the transition. These results further develop the notion of entanglement phase transitions in open systems, with direct implications for driven many-body localized systems, as well as experimental studies of driven-disordered systems.

DOI
10.1103/physrevlett.123.110601

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The Anderson localization transition is one of the most well studied examples of a zero temperature quantum phase transition. On the other hand, many open questions remain about the phenomenology of disordered systems driven far out of equilibrium. Here we study the localization transition in the prototypical three-dimensional, noninteracting Anderson model when the system is driven at its boundaries to induce a current carrying nonequilibrium steady state. Recently we showed that the diffusive phase of this model exhibits extensive mutual information of its nonequilibrium steady-state density matrix. We show that this extensive scaling persists in the entanglement and at the localization critical point, before crossing over to a short-range (area-law) scaling in the localized phase. We introduce an entanglement witness for fermionic states that we name the mutual coherence, which, for fermionic Gaussian states, is also a lower bound on the mutual information. Through a combination of analytical arguments and numerics, we determine the finite-size scaling of the mutual coherence across the transition. These results further develop the notion of entanglement phase transitions in open systems, with direct implications for driven many-body localized systems, as well as experimental studies of driven-disordered systems.
We model a one-dimensional current-driven interacting disordered system through a non-Hermitian Hamiltonian with asymmetric hopping and study the entanglement properties of its eigenstates. In particular, we investigate whether a many-body localizable system undergoes a transition to a current-carrying nonequilibrium steady state under the drive and how the entanglement properties of the quantum states change across the transition. We also discuss the dynamics, entanglement growth, and long-time fate of a generic initial state under an appropriate time evolution of the system governed by the non-Hermitian Hamiltonian. Our study reveals rich entanglement structures of the eigenstates of the non-Hermitian Hamiltonian. We find transition between current-carrying states with volume-law to area-law entanglement entropy, as a function of disorder and the strength of the non-Hermitian term.
Characterizing the delocalization transition in closed quantum systems with a many-body localized phase is a key open question in the field of nonequilibrium physics. We exploit the fact that localization of particles as realized in Anderson localization and standard many-body localization (MBL) implies Fock-space localization in single-particle basis sets characterized by a real-space index. Using a recently introduced quantitative measure for Fock-space localization computed from the density distributions, the occupation distance, we systematically study its scaling behavior across delocalization transitions and identify critical points from scaling collapses of numerical data. Excellent agreement with literature results is found for the critical disorder strengths of noninteracting fermions, such as the one-dimensional Aubry-Andr\'e model and the three-dimensional Anderson model. We observe a distinctively different scaling behavior in the case of interacting fermions with random disorder consistent with a Kosterlitz-Thouless transition. Finally, we use our measure to extract the transition point as a function of filling for interacting fermions.
The quest for nonequilibrium quantum phase transitions is often hampered by the tendency of driving and dissipation to give rise to an effective temperature, resulting in classical behavior. Could this be different when the dissipation is engineered to drive the system into a nontrivial quantum coherent steady state? In this work we shed light on this issue by studying the effect of disorder on recently introduced dissipation-induced Chern topological states, and examining the eigenmodes of the Hermitian steady-state density matrix or entanglement Hamiltonian. We find that, similarly to equilibrium, each Landau band has a single delocalized level near its center. However, using three different finite-size scaling methods we show that the critical exponent $\ensuremath{\nu}$ describing the divergence of the localization length upon approaching the delocalized state is significantly different from equilibrium if disorder is introduced into the nondissipative part of the dynamics. This indicates a different type of nonequilibrium quantum critical universality class accessible in cold-atom experiments.
We study the entanglement Hamiltonian (EH) associated with the reduced density matrix of free fermion models in a delocalized-localized Anderson phase transition. We show numerically that the structure of the EH matrix differentiates the delocalized from the localized phase. In the delocalized phase, EH becomes a long-range Hamiltonian but is short-range in the localized phase, no matter what the configuration of the system's Hamiltonian is (whether it is long- or short-range). With this view, we introduce the entanglement conductance (EC), which quantifies how much EH is long-range, and we propose it as an alternative quantity to measure entanglement in the Anderson phase transition, by which we locate the phase transition point of some one-dimensional free fermion models. In addition, by applying the finite-size method to the EC, we find three-dimensional Anderson phase transition critical disorder strength.
Existence of Anderson localization is considered a manifestation of coherence of classical and quantum waves in disordered systems. Signatures of localization have been observed in condensed matter and cold atomic systems where the coupling to the environment can be significantly suppressed but not eliminated. In this work we explore the phenomena of localization in random Lindbladian dynamics describing open quantum systems. We propose a model of one-dimensional chain of non-interacting, spinless fermions coupled to a local ensemble of baths. The jump operator mediating the interaction with the bath linked to each site has a power-law tail with an exponent ppp. We show that the steady state of the system undergoes a localization entanglement phase transition by tuning ppp which remains stable in the presence of coherent hopping. Unlike the entanglement transition in the quantum trajectories of open systems, this transition is exhibited by the averaged steady state density matrix of the Lindbladian. The steady state in the localized phase is characterised by a heterogeneity in local population imbalance, while the jump operators exhibit a constant participation ratio of the sites they affect. Our work provides a novel realisation of localization physics in open quantum systems.
The quest for nonequilibrium quantum phase transitions is often hampered by the tendency of driving and dissipation to give rise to an effective temperature, resulting in classical behavior. In this work we shed new light on this issue by studying the effect of disorder on recently-introduced dissipation-induced Chern topological states, and examining the eigenmodes of the Hermitian steady state density matrix or entanglement Hamiltonian. We find that, similarly to equilibrium, each Landau band has a single delocalized level near its center. However, using three different finite size scaling methods we show that the critical exponent ν\nuν describing the divergence of the localization length upon approaching the delocalized state is significantly different from equilibrium if disorder is introduced into the non-dissipative part of the dynamics. This indicates a new nonequilibrium quantum critical universality class accessible in cold-atom experiments.