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Localization in the random XXZ quantum spin chain

Alexander Elgart, Abel Klein

Abstract

We study the many-body localization (MBL) properties of the Heisenberg XXZ spin-$\frac12$ chain in a random magnetic field. We prove that the system exhibits localization in any given energy interval at the bottom of the spectrum in a nontrivial region of the parameter space. This region, which includes weak interaction and strong disorder regimes, is independent of the size of the system and depends only on the energy interval. Our approach is based on the reformulation of the localization problem as an expression of quasi-locality for functions of the random many-body XXZ Hamiltonian. This allows us to extend the fractional moment method for proving localization, previously derived in a single-particle localization context, to the many-body setting.

Localization in the random XXZ quantum spin chain

Abstract

We study the many-body localization (MBL) properties of the Heisenberg XXZ spin- chain in a random magnetic field. We prove that the system exhibits localization in any given energy interval at the bottom of the spectrum in a nontrivial region of the parameter space. This region, which includes weak interaction and strong disorder regimes, is independent of the size of the system and depends only on the energy interval. Our approach is based on the reformulation of the localization problem as an expression of quasi-locality for functions of the random many-body XXZ Hamiltonian. This allows us to extend the fractional moment method for proving localization, previously derived in a single-particle localization context, to the many-body setting.
Paper Structure (18 sections, 14 theorems, 172 equations)

This paper contains 18 sections, 14 theorems, 172 equations.

Table of Contents

  1. Introduction
  2. Model description and main results
  3. Model description
  4. Main results
  5. Key ingredients for the proofs
  6. Some definitions
  7. Quasi-locality for resolvents
  8. An a-priori estimate
  9. Dressing resolvents with Hilbert-Schmidt operators
  10. Large deviation estimate
  11. Decoupling of resolvents
  12. Clusters classification
  13. Decoupling revisited
  14. Reduction to lower energies
  15. Proof of the main theorem
  16. ...and 3 more sections

Key Result

Theorem 2.4

Fix $\Delta_0>1$, $\lambda_0 >0$, and let $s\in(0,\frac{1}{3})$. Then for all $k\in\mathbb N^0$ there exist constants $D_k,F_k,\xi_k, \theta_k>0$ (depending on $k$, $\Delta_0$, $\lambda_0$ and $s$) such that, for all $\Delta \ge \Delta_0$ and $\lambda \ge \lambda_0$ with $\lambda \Delta^2\ge D_k$, $ for $A\subset B\subset \Lambda$ with $A$ connected in $\Lambda$.

Theorems & Definitions (32)

  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Theorem 2.4: Quasi-locality for resolvents
  • Remark 2.5
  • Corollary 2.6: Quasi-locality for Borel functions
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Remark 3.3
  • ...and 22 more