• Open Access

Criterion for Many-Body Localization-Delocalization Phase Transition

Maksym Serbyn, Z. Papić, and Dmitry A. Abanin
Phys. Rev. X 5, 041047 – Published 23 December 2015

Abstract

We propose a new approach to probing ergodicity and its breakdown in one-dimensional quantum many-body systems based on their response to a local perturbation. We study the distribution of matrix elements of a local operator between the system’s eigenstates, finding a qualitatively different behavior in the many-body localized (MBL) and ergodic phases. To characterize how strongly a local perturbation modifies the eigenstates, we introduce the parameter G(L)=ln(Vnm/δ), which represents the disorder-averaged ratio of a typical matrix element of a local operator V to energy level spacing δ; this parameter is reminiscent of the Thouless conductance in the single-particle localization. We show that the parameter G(L) decreases with system size L in the MBL phase and grows in the ergodic phase. We surmise that the delocalization transition occurs when G(L) is independent of system size, G(L)=Gc1. We illustrate our approach by studying the many-body localization transition and resolving the many-body mobility edge in a disordered one-dimensional XXZ spin-1/2 chain using exact diagonalization and time-evolving block-decimation methods. Our criterion for the MBL transition gives insights into microscopic details of transition. Its direct physical consequences, in particular, logarithmically slow transport at the transition and extensive entanglement entropy of the eigenstates, are consistent with recent renormalization-group predictions.

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  • Received 19 July 2015

DOI:https://doi.org/10.1103/PhysRevX.5.041047

This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Authors & Affiliations

Maksym Serbyn1, Z. Papić2, and Dmitry A. Abanin3,4

  • 1Department of Physics, University of California, Berkeley, California 94720, USA
  • 2School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom
  • 3Department of Theoretical Physics, University of Geneva, 24 quai Ernest-Ansermet, 1211 Geneva, Switzerland
  • 4Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada

Popular Summary

Statistical physics describes ergodic systems—systems that will dynamically explore all available states and ultimately reach thermal equilibrium. Recent studies have shown that many-body localization breaks ergodicity, which leads to the absence of thermalization. Here, we introduce a method to probe ergodicity and many-body localization using the response of the system’s eigenstates to a local perturbation. Our study focuses on interacting lattice models of spins or fermions in the presence of quenched disorder.

We introduce a many-body generalization of the Thouless conductance—a quantity that plays a central role in the scaling theory of Anderson localization. Physically, our “many-body Thouless conductance” describes the sensitivity of the system’s eigenstates to a local perturbation. In the ergodic phase, a local perturbation mixes an extensive number of eigenstates. In contrast, the many-body localization phase has an extensive number of local integrals of motion, and its eigenstates are stable to local perturbations because those cannot alter the integrals of motion with support away from the perturbation site.

We detect the many-body localization-ergodic transition as the point where integrals of motion cease to be local. We formulate a delocalization criterion that is reminiscent of the Thouless criterion in single-particle systems. We use this criterion to demonstrate the existence of the “many-body mobility edge” in a disordered one-dimensional spin chain, and we obtain insights into the critical behavior at the many-body localization-ergodic transition.

We expect that our findings will pave the way to a microscopic scaling theory of the many-body localization-ergodic phase transition.

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Vol. 5, Iss. 4 — October - December 2015

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