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Localization of interacting random particles with power-law long-range hopping

Wenwen Jian, Yingte Sun

TL;DR

The article analyzes an $N$-particle lattice system with power-law long-range hopping and random potential, proving power-law localization under strong disorder. The authors develop and implement a long-range, multi-scale analysis of Green's functions, partitioning $n$-particle cubes into partially interactive and fully interactive classes to manage correlations. They establish a robust inductive framework that propagates non-resonant behavior across scales, culminating in a pure point spectrum and polynomial decay of eigenfunctions with explicit rates. The work extends single-particle power-law localization to multi-particle settings with interactions and long-range hopping, providing rigorous foundational results relevant to localization phenomena in disordered quantum systems. The methods rely on precise probabilistic (Stollmann/Wegner) bounds, coupling lemmas, and a carefully designed scale induction, highlighting the feasibility of multi-particle localization in the presence of long-range couplings.

Abstract

In this paper, we study the interacting random particles with power-law long-rang hopping. Via the multi-scale analysis arguments for the Green's function, we establish the power-law localization for all energy with strong disorder.

Localization of interacting random particles with power-law long-range hopping

TL;DR

The article analyzes an -particle lattice system with power-law long-range hopping and random potential, proving power-law localization under strong disorder. The authors develop and implement a long-range, multi-scale analysis of Green's functions, partitioning -particle cubes into partially interactive and fully interactive classes to manage correlations. They establish a robust inductive framework that propagates non-resonant behavior across scales, culminating in a pure point spectrum and polynomial decay of eigenfunctions with explicit rates. The work extends single-particle power-law localization to multi-particle settings with interactions and long-range hopping, providing rigorous foundational results relevant to localization phenomena in disordered quantum systems. The methods rely on precise probabilistic (Stollmann/Wegner) bounds, coupling lemmas, and a carefully designed scale induction, highlighting the feasibility of multi-particle localization in the presence of long-range couplings.

Abstract

In this paper, we study the interacting random particles with power-law long-rang hopping. Via the multi-scale analysis arguments for the Green's function, we establish the power-law localization for all energy with strong disorder.

Paper Structure

This paper contains 23 sections, 24 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Some Remarks
  4. Organization of the Paper
  5. Some Properties Of $n$-Particle Cubes
  6. Notations
  7. Separability of Cubes
  8. Singular and Non-singular $n$-particle Cubes
  9. The MSA Result.
  10. The Induction of Iteration Lemmas
  11. Analysis of Partially Interactive Pairs
  12. Analysis of Fully Interactive Pairs
  13. Analysis of Mixed Pairs
  14. The initialization of the MSA
  15. The conclusion of multi-particle multi-scale analysis
  16. ...and 8 more sections

Key Result

Theorem 1.1

Fix $N\geq 2$, and consider the $N$-particle random Hamiltonian $\textbf{H}^{(N)}_{\omega}$ given by (n). Suppose that Assumptions $\textbf{A}$, $\textbf{B}$ and $\textbf{C}$ hold true. Let Then there exists $g^*=g^*(N,d,\kappa,\rho,r,\mathrm{r}_0,M,M_1)\in(0,+\infty)$ such that for any $g$ with $|g|\geq g^*$, with $\mathbb{P}$-probability one, the operator $\textbf{H}^{(N)}_{\omega}$ is pure poi

Theorems & Definitions (51)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Lemma 2.4: Lemma 4.2.2 in CS2014
  • Lemma 2.5: Lemma 4.2.3 in CS2014
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 41 more