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Many-body localization for the random XXZ spin chain in fixed energy intervals

Alexander Elgart, Abel Klein

TL;DR

This work rigorously establishes slow information propagation, a hallmark of many-body localization, for the infinite random XXZ spin-$\tfrac12$ chain within any fixed energy interval near the bottom of the spectrum. By combining finite-volume MBL results with energy-window filtering and a careful reduction to effectively finite regions whose size scales with time, the authors prove a logarithmic light cone for energy-filtered dynamics: the energy-projected evolution can be approximated by a localized observable with error decaying exponentially in the confinement scale, up to a polynomial in time. The analysis hinges on three guiding principles—restricting particle configurations under energy windows, approximating energy indicators by Fourier-supported functions, and a finite-speed propagation bound—and culminates in an infinite-volume bound that is independent of system size. The results bridge zero-temperature localization and many-body localization, extending prior finite-volume and droplet-spectrum results to the full infinite-volume setting and providing rigorous evidence for MBL-like behavior in fixed-energy, low-density regimes of disordered quantum spin chains.

Abstract

A key signature of MBL (many-body localization) is the slow rate at which information spreads. It is shown that the infinite random Heisenberg XXZ spin-$\frac12$ chain exhibits slow propagation of information (logarithmic light cone) in any arbitrary but fixed energy interval. The relevant parameter regime, which covers both weak interaction and strong disorder, is determined solely by the energy interval.

Many-body localization for the random XXZ spin chain in fixed energy intervals

TL;DR

This work rigorously establishes slow information propagation, a hallmark of many-body localization, for the infinite random XXZ spin- chain within any fixed energy interval near the bottom of the spectrum. By combining finite-volume MBL results with energy-window filtering and a careful reduction to effectively finite regions whose size scales with time, the authors prove a logarithmic light cone for energy-filtered dynamics: the energy-projected evolution can be approximated by a localized observable with error decaying exponentially in the confinement scale, up to a polynomial in time. The analysis hinges on three guiding principles—restricting particle configurations under energy windows, approximating energy indicators by Fourier-supported functions, and a finite-speed propagation bound—and culminates in an infinite-volume bound that is independent of system size. The results bridge zero-temperature localization and many-body localization, extending prior finite-volume and droplet-spectrum results to the full infinite-volume setting and providing rigorous evidence for MBL-like behavior in fixed-energy, low-density regimes of disordered quantum spin chains.

Abstract

A key signature of MBL (many-body localization) is the slow rate at which information spreads. It is shown that the infinite random Heisenberg XXZ spin- chain exhibits slow propagation of information (logarithmic light cone) in any arbitrary but fixed energy interval. The relevant parameter regime, which covers both weak interaction and strong disorder, is determined solely by the energy interval.
Paper Structure (15 sections, 8 theorems, 83 equations)

This paper contains 15 sections, 8 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. The random XXZ spin chain and the main result
  3. Setting
  4. Logarithmic vs linear light cones
  5. Comparison with existing results
  6. Sketch of the proof of Theorem \ref{['thm:main']}
  7. The main result revisited
  8. Notation and definitions
  9. The XXZ Hamiltonian on subsets of Z
  10. Finite volume theorem and detailed main result
  11. Guiding principles for the proof of the main result
  12. Restriction on the location of particles
  13. Approximation of an energy interval indicator
  14. Finite speed of propagation
  15. Proof of the main result

Key Result

Theorem 2.1

Fix parameters $\Delta_0>1$ and $\lambda_0 >0$. Then for all $E\ge 0$ there exist strictly positive constants $C_E, D_E, \kappa_E, m_E$ (depending on $E$, $\Delta_0$, $\lambda_0$) such that for all $\Delta \ge \Delta_0$ and $\lambda \ge \lambda_0$ with $\lambda \Delta^2\ge D_E$ the following assert

Theorems & Definitions (14)

  • Theorem 2.1: Slow propagation of information-infinite volume
  • Theorem 3.1: Slow propagation of information-finite volume
  • Remark 3.2
  • Theorem 3.3: Slow propagation of information-infinite volume, detailed version
  • Proposition 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • ...and 4 more