Quantum Diffusion and Eigenfunction Delocalization in a Random Band Matrix Model

TL;DR

This paper demonstrates that quantum particles in random band matrices exhibit diffusive behavior over certain time scales and that most eigenvectors are delocalized, with localization lengths scaling with the band width.

Contribution

It provides rigorous proofs of quantum diffusion and eigenfunction delocalization in high-dimensional random band matrices, extending understanding of quantum dynamics in disordered systems.

Findings

Quantum diffusion occurs for times t << W^{d/3}.

Most eigenvectors have localization lengths exceeding W^{d/6} times the band width.

Results are uniform across matrix sizes.

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, uniformly distributed random variables if $\abs{x-y}$ is less than the band width $W$, and zero otherwise. 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 an arbitrarily large majority of the eigenvectors is larger than a factor $W^{d/6}$ times the band width. All results are uniform in the size $\abs{\Lambda}$ of the matrix.