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Slow propagation of information on the random XXZ quantum spin chain

Alexander Elgart, Abel Klein

TL;DR

This work establishes a rigorous link between localization in a fixed energy interval and slow information propagation in the random XXZ spin chain. Building on the authors' prior quasi-locality result (EK22), it proves that the finite-volume dynamics projected to a fixed energy window cannot propagate information quickly: observables supported on a region $\mathcal X$ have an evolved approximation $T_t$ whose support grows only modestly with time, with errors decaying exponentially in the extension parameter $\ell$. The analysis combines deterministic quasi-locality, decoupling, and Helffer–Sjöstrand functional calculus to yield explicit bounds involving polynomial time factors and exponential spatial decay, and further provides a matrix-element version that mitigates volume dependence to logarithmic levels. These results reinforce the rigorous understanding of MBL-like behavior by linking quasi-local spectral properties to dynamical information spreading, at least within finite energy windows. They also offer a tractable, interval-restricted framework that could inform studies of infinite-volume limits and related localization phenomena in disordered quantum spin systems.

Abstract

The random XXZ quantum spin chain manifests localization (in the form of quasi-locality) in any fixed energy interval, as previously proved by the authors. In this article it is shown that this property implies slow propagation of information, one of the putative signatures of many-body localization, in the same energy interval.

Slow propagation of information on the random XXZ quantum spin chain

TL;DR

This work establishes a rigorous link between localization in a fixed energy interval and slow information propagation in the random XXZ spin chain. Building on the authors' prior quasi-locality result (EK22), it proves that the finite-volume dynamics projected to a fixed energy window cannot propagate information quickly: observables supported on a region have an evolved approximation whose support grows only modestly with time, with errors decaying exponentially in the extension parameter . The analysis combines deterministic quasi-locality, decoupling, and Helffer–Sjöstrand functional calculus to yield explicit bounds involving polynomial time factors and exponential spatial decay, and further provides a matrix-element version that mitigates volume dependence to logarithmic levels. These results reinforce the rigorous understanding of MBL-like behavior by linking quasi-local spectral properties to dynamical information spreading, at least within finite energy windows. They also offer a tractable, interval-restricted framework that could inform studies of infinite-volume limits and related localization phenomena in disordered quantum spin systems.

Abstract

The random XXZ quantum spin chain manifests localization (in the form of quasi-locality) in any fixed energy interval, as previously proved by the authors. In this article it is shown that this property implies slow propagation of information, one of the putative signatures of many-body localization, in the same energy interval.
Paper Structure (12 sections, 10 theorems, 130 equations)

This paper contains 12 sections, 10 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. The model, localization, and the main result
  3. Model description
  4. Localization as quasi-locality
  5. Slow propagation of information
  6. Key proof ingredients
  7. Preliminaries
  8. Quasi-locality (deterministic)
  9. Consequences of quasi-locality
  10. Decoupling
  11. Proof of slow propagation of information
  12. Proof of slow propagation of information, matrix elements version

Key Result

Theorem 1

For a given energy $E>0$, there exists a non-trivial region in the $(\Delta,\lambda)$ parameter space, such that for any fixed point in this region, scales $L,\ell\in\mathbb N$, and all $t\in\mathbb R$, the following holds: For every observable $\mathcal{O}$ supported on a discrete interval $[a,b] \ where $\mathbb E$ stands for the expectation with respect to $\omega$, $P_{[0,E]}$ is the spectral

Theorems & Definitions (22)

  • Theorem : Slow propagation of information, informal
  • Definition 2.1
  • Theorem 2.2: Quasi-locality
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Theorem 2.6: Slow propagation of information
  • Corollary 2.7: Slow propagation of information, matrix elements version
  • Lemma 3.1: EK22
  • Lemma 3.2
  • ...and 12 more