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Localization phenomena in the random XXZ spin chain

Alexander Elgart, Abel Klein

TL;DR

This work proves localization phenomena for the infinite random XXZ spin-1/2 chain in a fixed energy interval, establishing spectral, eigenstate, and weak dynamical localization under a large $\lambda\Delta^2$ regime. The authors develop a finite-volume criterion for Green function decay and show it implies uniform infinite-volume decay, enabling localization results uniformly in particle number. The analysis integrates Combes-Thomas bounds, resolvent identities, and large-deviation estimates to overcome the many-body complexity and the scarcity of randomness. Consequently, the paper provides rigorous evidence of an MBL-like phase in a non-exactly solvable infinite spin system, in the zero-temperature, fixed-energy setting, and connects localization to dynamical constraints via eigencorrelator decay and a quasi-local structure.

Abstract

It is shown that the infinite random Heisenberg XXZ spin-$\frac12$ chain exhibits localization phenomena, such as spectral, eigenstate, and weak dynamical localization, in an arbitrary (but fixed) energy interval in a non-trivial region of the parameter space. This region depends only on the energy interval and includes weak interaction and strong disorder regimes. The crucial step in the argument is a proof that if the Green functions for the associated finite systems Hamiltonians exhibit certain (volume-dependent) decay properties in a fixed energy interval, then the infinite volume Green function decays in the same interval as well. The pertinent finite systems decay properties for the random XXZ spin chain had been previously verified by the authors.

Localization phenomena in the random XXZ spin chain

TL;DR

This work proves localization phenomena for the infinite random XXZ spin-1/2 chain in a fixed energy interval, establishing spectral, eigenstate, and weak dynamical localization under a large regime. The authors develop a finite-volume criterion for Green function decay and show it implies uniform infinite-volume decay, enabling localization results uniformly in particle number. The analysis integrates Combes-Thomas bounds, resolvent identities, and large-deviation estimates to overcome the many-body complexity and the scarcity of randomness. Consequently, the paper provides rigorous evidence of an MBL-like phase in a non-exactly solvable infinite spin system, in the zero-temperature, fixed-energy setting, and connects localization to dynamical constraints via eigencorrelator decay and a quasi-local structure.

Abstract

It is shown that the infinite random Heisenberg XXZ spin- chain exhibits localization phenomena, such as spectral, eigenstate, and weak dynamical localization, in an arbitrary (but fixed) energy interval in a non-trivial region of the parameter space. This region depends only on the energy interval and includes weak interaction and strong disorder regimes. The crucial step in the argument is a proof that if the Green functions for the associated finite systems Hamiltonians exhibit certain (volume-dependent) decay properties in a fixed energy interval, then the infinite volume Green function decays in the same interval as well. The pertinent finite systems decay properties for the random XXZ spin chain had been previously verified by the authors.

Paper Structure

This paper contains 10 sections, 13 theorems, 171 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The model and (informal) main result
  3. Relation with existing research results and open problems
  4. Main result and technical steps
  5. Basic features of the XXZ spin chain
  6. Proof of Theorem \ref{['cor:weakinf']}
  7. Proof of Theorem \ref{['thmdynloc']}
  8. Localization types and nomenclature
  9. Exponential sums
  10. Quasi-locality of the filter function

Key Result

Theorem 1.1

Let $H^\mathbb Z$ be the random XXZ Hamiltonian on $\mathcal{H}_\mathbb Z$ with parameters $\Delta>1$and $\lambda >0$. Fix the energy interval $I(E_0)=[0,E_0]$, where $E_0>0$. Then, if $\lambda \Delta^2$ is sufficiently large, we have:

Figures (1)

  • Figure 1: A localization cartoon for the infinite volume XXZ model in strong disorder/weak interaction regimes. The blue region $A$ is the few-particles localization $N\in [0,N_0]$, the green region $B$ is the zero temperature localization $E\in[0, E_0]$ (our result). The total region of localization can be extended to include the pink sector $C$ using existing methods. The white region $D$ is currently not understood.

Theorems & Definitions (27)

  • Theorem 1.1: Informal formulation
  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Theorem 2.4: Finite volumes criterion
  • Remark 2.5
  • Theorem 2.6
  • Remark 3.1
  • Lemma 4.1
  • ...and 17 more