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Self-consistent equations and quantum diffusion for the Anderson model

Adam Black, Reuben Drogin, Felipe Hernández

Abstract

We consider the Anderson tight-binding model on $\mathbb{Z}^d$, $d\geq 2$, with Gaussian noise and at low disorder $λ>0$. We derive a diffusive scaling limit for the entries of the resolvent $R(z)$ at imaginary part $\operatorname*{Im} z\simλ^{2+κ_d}$, $κ_d>0$, with high probability. As consequences, we establish quantum diffusion (in a time-averaged sense) for the Schrödinger propagator at the longest timescale known to date and improve the best available lower bounds on the localization length of eigenfunctions. Our results for $d=2$ are the first quantum diffusion results for the Anderson model on $\mathbb{Z}^2$. The proof avoids the use of diagrammatic expansions and instead proceeds by analyzing certain self-consistent equations for $R(z)$. This is facilitated by new estimates for $\|R(z)\|_{\ell^p\rightarrow \ell^q}$ that control the recollisions.

Self-consistent equations and quantum diffusion for the Anderson model

Abstract

We consider the Anderson tight-binding model on , , with Gaussian noise and at low disorder . We derive a diffusive scaling limit for the entries of the resolvent at imaginary part , , with high probability. As consequences, we establish quantum diffusion (in a time-averaged sense) for the Schrödinger propagator at the longest timescale known to date and improve the best available lower bounds on the localization length of eigenfunctions. Our results for are the first quantum diffusion results for the Anderson model on . The proof avoids the use of diagrammatic expansions and instead proceeds by analyzing certain self-consistent equations for . This is facilitated by new estimates for that control the recollisions.

Paper Structure

This paper contains 37 sections, 33 theorems, 310 equations.

Table of Contents

  1. Introduction
  2. Main ideas and discussion
  3. Step 1: Concentration inequalities for the resolvent.
  4. Step 2: The self-consistent equation for ${\mathbb E} R$
  5. Step 3: The diffusive profile of $R$
  6. Diagrammatics and the $\|R\|_{1\to 4}$ estimate.
  7. Formal perturbation theory and renormalization
  8. Resolvent estimates as crossing integrals
  9. As a dispersive estimate
  10. Comparison to classical models
  11. Preliminaries and Conventions
  12. The Laplacian and the Fourier transform
  13. Density of states and the self-consistent equation
  14. Truncation to ${\mathbb{Z}}^d_L$
  15. Constants and asymptotic notation
  16. ...and 22 more sections

Key Result

Theorem 1.1

Let $\delta>0$ and $H$ be as in eq:tbm. There exists $c_{\delta},C_{\delta}>0$, independent of $\lambda$, such that for any $C_{\delta}\leq T\leq \lambda^{-2-\kappa_d+\delta}$ we have and with probability at least $1-C_{\delta}\lambda^{1000}$ and $\kappa_d>0$ given by

Theorems & Definitions (56)

  • Theorem 1.1: Dynamical delocalization
  • Theorem 1.2
  • Theorem 1.3: Delocalization bound for most eigenfunctions
  • Proposition 1.4
  • Proposition 2.1
  • proof
  • Proposition 3.1: A priori resolvent bounds
  • Proposition 3.2: Corollary 1.3 of BDH
  • Lemma 3.3
  • proof
  • ...and 46 more