Quantum Diffusion and Eigenfunction Delocalization in a Random Band Matrix Model

• Published in: arXiv.org
• Main Authors: Erdos, Laszlo, Knowles, Antti
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 20.09.2010
• Subjects: Eigenvectors; Independent variables; Mathematics - Mathematical Physics; Mathematics - Probability; Matrix methods; Physics - Mathematical Physics; Random variables
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1002.1695

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.