Quantum Diffusion and Eigenfunction Delocalization in a Random Band Matrix Model

Details Quantum Diffusion and Eigenfunction Delocalization in a Random Band Matrix Model Quantum Diffusion and Eigenfunction Delocalization in a Random Band Matrix Model

2010-02-08

arxiv.org/abs/1002.1695v5

Physics

Mathematical Physics

Minor corrections, Sections 4 and 11 updated

Scientific paper

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.

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