On the localization length of finite-volume random block Schrödinger operators

Abstract

We study a general class of random block Schrödinger operators (RBSOs) in dimensions 1 and 2, which naturally extend the Anderson model by replacing the random potential with a random block potential. Specifically, we focus on two RBSOs -- the block Anderson and Wegner orbital models -- defined on the $d$-dimensional torus $(\mathbb{Z}/L\mathbb{Z}{)}^d$. They take the form $H=V+\lambda \Psi$, where $V$ is a block potential with i.i.d. $W^d\times W^d$ Gaussian diagonal blocks, $\Psi$ describes interactions between neighboring blocks, and $\lambda >0$ is a coupling parameter. We normalize the blocks of $\Psi$ so that each block has a Hilbert-Schmidt norm of the same order as the blocks of $V$. Assuming $W\ge L^{\delta }$ for a small constant $\delta >0$ and $\lambda \gg W^{-d/2}$, we establish the following results. In dimension $d=2$, we prove delocalization and quantum unique ergodicity for bulk eigenvectors. Combined with the localization result from arXiv:1608.02922, which holds under the condition $\lambda \ll W^{-d/2}$, this provides a rigorous proof of the Anderson localization-delocalization transition as $\lambda$ crosses the critical threshold $W^{-d/2}$. In dimension $d=1$, we show that the localization length of bulk eigenvectors is at least of order $(W\lambda {)}^2$, which is believed to be the correct scaling.

82 pages. We have relaxed several key assumptions and relocated some of the proofs to the appendix