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Recent progress in many-body localization

Dmitry A. Abanin, Zlatko Papić

arXiv:1705.09103v1cond-mat.dis-nncond-mat.mes-hallcond-mat.stat-mech

TL;DR

The article introduces many-body localization as a framework for understanding non-equilibrium dynamics in interacting, disordered quantum systems and surveys recent progress. It explains MBL through emergent LIOMs, discusses entanglement and quantum-correlation dynamics, reviews experiments, and identifies unresolved questions about transitions, disorder, and environmental effects.

  • Problem

    The paper addresses how generic interacting quantum systems evolve after quenches, including whether they thermalize, retain initial-state memory, or realize other non-thermalizing phases.

  • Method

    The article provides a brief synthesis of MBL theory, centered on LIOMs, dynamical properties, experiments, and open problems.

  • Results

    The review highlights that MBL retains initial-state memory, supports spreading entanglement, and exhibits logarithmic propagation of quantum correlations.

  • Takeaways & Limitations

    MBL offers a mechanism for avoiding thermalization and may support symmetry-related topological order and more robust quantum information processing.

Abstract

from arXiv · show

This article is a brief introduction to the rapidly evolving field of many-body localization. Rather than giving an in-depth review of the subject, our aspiration here is simply to introduce the problem and its general context, outlining a few directions where notable progress has been achieved in recent years. We hope that this will prepare the readers for the more specialized articles appearing in the forthcoming dedicated volume of Annalen der Physik, where these developments are discussed in more detail.

I. INTRODUCTION

Many-body localization studies how interactions affect Anderson localization and how isolated interacting quantum systems evolve after a quench. The article introduces MBL, its LIOM description, dynamical properties, experimental probes, and open questions.

  • Motivation: MBL research examines whether interactions destabilize the Anderson insulator and seeks to classify the non-equilibrium behavior of generic interacting, disordered quantum systems.Anderson insulators lack diffusion in low-dimensional disordered systems.
  • Quantum quenches: Quantum quenches prepare simple, often non-entangled states that undergo unitary evolution, allowing equilibration and long-time behavior to be studied experimentally.Such initial states are typically far from equilibrium and can be directly implemented in synthetic systems.
  • Many-body localization: Unlike thermalizing systems, MBL retains memory of the initial state because strong quenched disorder makes energy-exchange processes off-resonant.This breakdown of ergodicity prevents description by conventional statistical mechanics.
  • Many-body localization: A complete set of emergent quasi-local integrals of motion, called LIOMs or l-bits, provides a universal description of MBL phases and their lack of thermalization.The article presents LIOMs as a new kind of integrability.
  • Dynamics: Despite absent energy transport, entanglement spreads through MBL systems, producing equilibration to a highly non-thermal state.The LIOM picture accounts for both entanglement properties and its spreading.
  • Open questions: The MBL–ergodic transition lacks a complete theory, while rare-region effects and the stability of MBL under dissipation remain important open issues.The article also highlights non-ergodic states without quenched disorder and Floquet phases as directions not covered in depth.
  • Symmetry and applications: Symmetries constrain whether MBL can occur but may also stabilize finite-energy-density topological order and protect quantum information at higher temperatures.These possibilities motivate applications of MBL to quantum information processing.
  • Experiments: Experiments in ultracold atoms, trapped ions, and NV-center spins have provided complementary insights into MBL and thermalization dynamics.The review treats these experiments briefly and refers readers to the original articles.

II. EIGENSTATE THERMALIZATION HYPOTHESIS AND THE STRUCTURE OF ERGODIC EIGENSTATES

The ETH framework connects the structure of highly excited eigenstates to thermalization and entanglement in ergodic systems. Its matrix-element ansatz describes equilibrium values, fluctuations, relaxation, and volume-law entanglement.

  • Eigenstate structure: Non-equilibrium dynamics can be determined from highly excited eigenstates and their energies, but this description does not by itself transparently explain the resulting behavior.In a quench, the initial state is expanded in energy eigenstates whose phases govern time evolution.
  • ETH: The eigenstate thermalization hypothesis states that individual excited eigenstates have thermal expectation values matching microcanonical and Gibbs ensembles.This provides an explanation for thermalization in isolated ergodic systems.
  • ETH ansatz: The ETH ansatz separates operator matrix elements into a smooth diagonal term and a random off-diagonal term controlled by energy and frequency-dependent functions.The off-diagonal function f(E, ω) determines relaxation of physical observables.
  • Relaxation: The ETH ansatz predicts that local observables approach an equilibrium value with temporal fluctuations of order O(e^-S), independently of the initial state.Conditional fluctuations around the relaxation trajectory also have variance O(e^-S).
  • Entanglement and subsystems: For a sufficiently small subsystem, ETH makes its observables thermal because the complementary subsystem acts as an efficient thermal bath.The effective temperature is determined by the energy of the eigenstate.
  • Entanglement: In ETH eigenstates, entanglement entropy equals the subsystem’s thermodynamic entropy and generally scales with the volume of that subsystem.This reflects the high entanglement of ergodic eigenstates.

III. FROM SINGLE-PARTICLE TO MANY-BODY LOCALIZATION

Many-body localization asks whether localization survives when interactions and finite particle density are added to disordered systems. The article presents interacting one-dimensional XXZ and spinless-fermion models as central settings for studying this question.

  • Anderson localization: Anderson localization produces exponentially localized single-particle wave functions and eliminates diffusion at sufficiently strong disorder.In one and two dimensions, all states are localized even for weak disorder; in three dimensions, a mobility edge can separate localized and extended states.
  • From single-particle to many-body localization: The central open question is whether localization survives generic two-body interactions at finite particle density.This question motivates the transition from single-particle localization to many-body localization.
  • Interacting models: The commonly studied model is a one-dimensional spin-1/2 XXZ chain with nearest-neighbor hopping, interactions, and a uniformly distributed random field.The field strength is typically sampled as h_i ∈ [−W, W], with open-chain boundary conditions.
  • Interacting models: The XXZ chain maps through the Jordan-Wigner transform to interacting spinless fermions in a disordered one-dimensional crystal.The random field becomes an on-site chemical potential, while spin interactions become nearest-neighbor density-density interactions.
  • Evidence for MBL: Numerical studies establish that all states of the finite XXZ model become many-body localized when disorder exceeds a critical strength W > Wc.Finite lattice models have bounded spectra, enabling microscopic investigations through exact diagonalization of small one-dimensional systems.

IV. MANY-BODY LOCALIZATION: LOCAL INTEGRALS OF MOTION

The LIOM picture describes the MBL phase through quasi-local conserved operators that label eigenstates and preserve local information. Their existence is supported by perturbative, numerical, and mathematical studies.

  • Definition and structure: LIOMs are mutually commuting Pauli operators that commute with the Hamiltonian and have support decaying exponentially away from their associated sites.They are also called l-bits and have eigenvalues ±1.
  • Definition and structure: The simultaneous eigenvalues of all LIOMs uniquely specify each MBL eigenstate.Thus, the LIOMs form a complete set of conserved quantum numbers for the eigenstates.
  • Construction of LIOMs: At strong disorder, MBL eigenstates are related to product states by a quasi-local unitary transformation U.The transformation can be constructed perturbatively in λ = J/W ≪ 1 and creates predominantly nearby spin flips.
  • Physical consequences: Conservation of each quasi-local LIOM preserves local memory of the initial state at arbitrarily long times and explains ergodicity breaking.This follows because the expectation value of every τ^z_i remains conserved under unitary evolution.
  • Physical consequences: The LIOM framework explains entanglement properties and entanglement spreading after quantum quenches in MBL systems.The picture has been supported by explicit LIOM constructions, perturbative analyses, and a mathematical proof of quasi-locality under stated spectral assumptions.

V. ENTANGLEMENT AND CLASSICAL SIMULATIONS OF MANY-BODY LOCALIZED STATES

MBL eigenstates have low, boundary-law entanglement despite being highly excited, making them more tractable for classical simulation. Tensor-network methods exploit this structure, while the entanglement spectrum provides additional information.

  • Entanglement scaling: In one-dimensional MBL systems, eigenstate entanglement entropy saturates to a constant once the chain length exceeds the localization length.This behavior has been verified numerically in several one-dimensional models.
  • Entanglement scaling: In higher dimensions, MBL eigenstates generally obey boundary-law entanglement scaling proportional to the boundary degrees of freedom.This contrasts with the volume-law entanglement of excited eigenstates in ergodic systems.
  • Entanglement scaling: The boundary law persists for arbitrarily highly excited MBL eigenstates because of LIOMs, even though many-body level spacings vanish exponentially with system size.The cited spin-1/2 example has level splitting scaling as ∼W L/2^L.
  • Classical simulation: Low entanglement allows MBL eigenstates to be represented compactly and simulated efficiently with tensor-network methods.The required number of parameters scales polynomially with the number of degrees of freedom.
  • Classical simulation: Extensions of DMRG and spectral tensor-network approaches have been developed to obtain individual or complete sets of highly excited MBL eigenstates.These methods use the low-entanglement structure of MBL states.
  • Entanglement spectrum: The entanglement spectrum reveals information beyond entanglement entropy and differs in structure between MBL and ergodic systems.Ergodic states were found to obey the Marchenko-Pastur distribution in the cited studies.

VI. DYNAMICS IN MANY-BODY LOCALIZED PHASES

LIOMs provide a simple description of MBL dynamics: conserved effective spins interact through exponentially decaying couplings, producing slow dephasing, logarithmic correlation and entanglement spreading, and equilibration to a non-thermal state.

  • VI. DYNAMICS IN MANY-BODY LOCALIZED PHASES: LIOMs underpin MBL dynamics because each effective spin’s conserved z component precesses in a field determined by the other spins.A generic initial superposition therefore generates entanglement between remote effective spins.
  • VI. DYNAMICS IN MANY-BODY LOCALIZED PHASES: Exponentially decaying couplings between remote LIOMs make the entanglement time grow exponentially with separation.The characteristic length scale ˜ξ controls the dephasing dynamics.
  • VI. DYNAMICS IN MANY-BODY LOCALIZED PHASES: Quantum correlations propagate logarithmically in time, r(t) ∝ln(J0t/ℏ), rather than linearly as in ergodic systems.This behavior follows from the slow dephasing generated by interactions between remote effective spins.
  • VI. DYNAMICS IN MANY-BODY LOCALIZED PHASES: Logarithmic entanglement spreading is a characteristic MBL signature after a product-state quench.At long times, subsystem entanglement becomes extensive but is typically smaller than the full thermal entropy.
  • VI. DYNAMICS IN MANY-BODY LOCALIZED PHASES: Random phases equilibrate all local observables through a power-law approach, while the resulting state remains highly non-thermal and retains initial-condition memory.This dephasing mechanism distinguishes interacting MBL from the non-interacting Anderson insulator, where no quench equilibration occurs.

VII. MANY-BODY LOCALIZATION TRANSITION AND GRIFFITHS EFFECTS

The MBL transition is a dynamical transition from localized to ergodic eigenstates, with unresolved microscopic theory and important rare-region effects near and across the transition.

  • VII. MANY-BODY LOCALIZATION TRANSITION AND GRIFFITHS EFFECTS: Reducing disorder drives a transition in which eigenstate entanglement changes from boundary-law to volume-law scaling.Near the transition, localized and thermalizing subsystems coexist, and thermal regions can thermalize localized regions with finite-size limitations.
  • VII. MANY-BODY LOCALIZATION TRANSITION AND GRIFFITHS EFFECTS: A possible many-body Thouless conductance has been proposed as a parameter describing eigenstate responses to local Hamiltonian perturbations near the transition.Whether the transition obeys single-parameter scaling remains an open question requiring further investigation.
  • VII. MANY-BODY LOCALIZATION TRANSITION AND GRIFFITHS EFFECTS: A complete microscopic theory of the MBL transition is currently lacking.The article therefore discusses this subject only briefly.
  • VII. MANY-BODY LOCALIZATION TRANSITION AND GRIFFITHS EFFECTS: Phenomenological real-space renormalization-group rules model the transition by merging thermal and MBL subsystems, capturing their competition.Rare-region effects become important on both sides of the transition and determine various physical properties.

VIII. SYMMETRIES AND LOCALIZATION-PROTECTED QUANTUM ORDER

Symmetries strongly constrain whether MBL exists and what ordered phases it can support, including localization-protected quantum order at finite energy density.

  • VIII. SYMMETRIES AND LOCALIZATION-PROTECTED QUANTUM ORDER: The existence and character of MBL eigenstates depend strongly on the symmetries present in the system.Energy and total-magnetization conservation in models such as XXZ can be broken without destroying MBL, so these symmetries are not essential in that setting.
  • VIII. SYMMETRIES AND LOCALIZATION-PROTECTED QUANTUM ORDER: MBL eigenstates can support ordering even when the corresponding thermodynamic equilibrium is unordered because they are non-thermal.This creates the possibility of symmetry-breaking and topological order at finite energy density.
  • VIII. SYMMETRIES AND LOCALIZATION-PROTECTED QUANTUM ORDER: MBL can protect certain kinds of topological and symmetry-protected topological order at finite energy density.The possibility depends on the symmetry constraints compatible with the MBL phase.
  • VIII. SYMMETRIES AND LOCALIZATION-PROTECTED QUANTUM ORDER: Abelian symmetries allow either symmetric or symmetry-breaking MBL phases, whereas discrete non-Abelian symmetries require eigenstates to spontaneously break the symmetry.Continuous non-Abelian symmetries appear to prohibit MBL altogether because degeneracies generate resonances and violate boundary-law entanglement scaling.

IX. EXPERIMENTAL DEVELOPMENTS

Synthetic quantum systems provide controlled platforms for probing MBL dynamics, and experiments have observed MBL signatures in ultracold atoms, trapped ions, and related spin systems. Environmental coupling remains a central experimental limitation because slow extrinsic processes can destroy MBL.

  • IX. EXPERIMENTAL DEVELOPMENTS: Experiments have observed MBL signatures in one- and two-dimensional ultracold atoms in disordered optical lattices and in small trapped-ion systems.These experiments implement quantum quenches using prepared initial charge-density-wave states.
  • IX. EXPERIMENTAL DEVELOPMENTS: Synthetic systems enable controlled studies of logarithmic entanglement spreading, local-observable equilibration, symmetry effects, and symmetry-protected order.Their tunability supports investigations beyond what exact diagonalization can treat numerically in corresponding model systems.
  • IX. EXPERIMENTAL DEVELOPMENTS: Synthetic systems are not fully isolated, and slow extrinsic processes affect their states and typically destroy MBL.In ultracold-atom systems, these processes are sufficiently slow for clear MBL signatures to be observed, but they limit access to intrinsically slow dynamics.

X. OUTLOOK AND SOME OPEN QUESTIONS

Despite a largely complete picture of MBL at sufficiently strong disorder, the MBL–thermal transition and the stability of localization beyond the standard setting remain unresolved. Open directions include higher dimensions, baths, translation-invariant systems, other non-thermalizing phases, and periodically driven systems.

  • MBL–thermal transition: A complete theory of the MBL–thermal transition is still lacking, especially a microscopic account of finite thermalizing regions adjacent to MBL subsystems.This issue is also relevant to understanding MBL stability in higher dimensions.
  • Stability beyond one dimension: The stability of MBL in two dimensions and in the presence of long-range dipolar interactions or a bath remains experimentally relevant but theoretically unsettled.Cold-atom experiments in two dimensions are limited by short timescales, while interpreting NV-center experiments requires understanding dipolar interactions.
  • Other non-thermalizing phases: Other possible non-thermalizing phases could involve partial sets of quasi-local integrals of motion or delocalized but non-ergodic states, although some proposed examples remain disputed.The random-regular-graph hopping problem is non-local, whereas locality appears crucial to MBL.
  • Translation-invariant models: Whether quenched disorder is necessary for ergodicity breaking remains open because translation-invariant models may delocalize and thermalize through rare mobile resonant bubbles.These models use predominantly off-resonant transitions, making their basic mechanism similar to MBL.
  • Translation-invariant models: Numerical studies of translation-invariant models are difficult because severe finite-size effects obscure their behavior.More recent work investigates non-generic models that map disorder to an ancillary degree of freedom.
  • Floquet developments: Disordered Floquet systems can support Floquet-MBL phases that avoid indefinite heating and retain LIOMs and slow entanglement spreading.Floquet systems have periodically varying Hamiltonians and generally absorb energy toward an infinite-temperature state, making this non-thermalizing behavior distinctive.
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