Quantum Diffusion and Delocalization for Band Matrices with General Distribution

TL;DR

This paper proves diffusive quantum dynamics and delocalization properties for general random band matrices, extending previous results to broader classes with subexponential decay and symmetric distributions.

Contribution

It extends existing results on quantum diffusion and eigenvector delocalization to more general band matrices with arbitrary symmetric distributions.

Findings

Quantum particles exhibit diffusive behavior on specific time scales.

Eigenvectors have localization lengths larger than the band width by a factor of W^{d/6}.

Largest eigenvalue is bounded near 2 with high probability.

Abstract

We consider Hermitian and symmetric random band matrices $H$ in $d \geq 1$ dimensions. The matrix elements $H_{xy}$, indexed by $x,y \in \Lambda \subset \Z^d$, are independent and their variances satisfy $\sigma_{xy}^2:=\E \abs{H_{xy}}^2 = W^{-d} f((x - y)/W)$ for some probability density $f$. We assume that the law of each matrix element $H_{xy}$ is symmetric and exhibits subexponential decay. We prove that the time evolution of a quantum particle subject to the Hamiltonian $H$ is diffusive on time scales $t\ll W^{d/3}$. We also show that the localization length of the eigenvectors of $H$ is larger than a factor $W^{d/6}$ times the band width $W$. All results are uniform in the size $\abs{\Lambda}$ of the matrix. This extends our recent result \cite{erdosknowles} to general band matrices. As another consequence of our proof we show that, for a larger class of random matrices satisfying $\sum_x\sigma_{xy}^2=1$ for all $y$, the largest eigenvalue of $H$ is bounded with high probability by $2 + M^{-2/3 + \epsilon}$ for any $\epsilon > 0$, where $M \deq 1 / (\max_{x,y} \sigma_{xy}^2)$.