Quantum Diffusion and Delocalization for Band Matrices with General Distribution

Abstract

We consider Hermitian and symmetric random band matrices H in \({d \geqslant 1}\) dimensions. The matrix elements H xy , indexed by \({x,y \in \Lambda \subset \mathbb{Z}^d}\), are independent and their variances satisfy \({\sigma_{xy}^2:=\mathbb{E} |{H_{xy}}|^2 = W^{-d} f((x - y)/W)}\) 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 |Λ| of the matrix. This extends our recent result (Erdős and Knowles in Commun. Math. Phys., 2011) 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 \({2 + M^{-2/3 + \varepsilon}}\) for any \({\varepsilon > 0}\), where \({M := 1 / (\max_{x,y}\sigma_{xy}^2)}\) .