Delocalization of a general class of random block Schrödinger operators

Fan Yang, Jun Yin

Published 2025-01-15Version 1

We consider a natural class of extensions of the Anderson model on $\mathbb Z^d$, called random block Schr\"odinger operators (RBSO), which feature a random block potential. They are defined on the $d$-dimensional lattice $(\mathbb Z/L\mathbb Z)^d$ and take the form $H=V+\lambda \Psi$, where $V$ is a diagonal block matrix with diagonal blocks that are i.i.d. $W^d\times W^d$ GUE, $\Psi$ describes the interactions between neighboring blocks, and $\lambda>0$ is a coupling parameter. We focus on three specific RBSOs: the block Anderson model, where $\Psi$ is the Laplacian operator on $\mathbb Z^d$; the Anderson orbital model, where $\Psi$ is a block Laplacian operator; the Wegner orbital model, where the nearest-neighbor blocks of $\Psi$ are random matrices. Let $\Lambda_\Psi$ denote the strength of interaction between two neighboring blocks. Assuming $d\ge 7$ and $W\ge L^\epsilon$ for a small constant $\epsilon>0$, under the condition $ W^{d/4}/\Lambda_\Psi\ll\lambda\ll 1$, we prove delocalization and quantum unique ergodicity for the bulk eigenvectors, as well as a quantum diffusion estimate for the Green's function. Together with the localization result proved in arXiv:1608.02922, our results rigorously establish an Anderson localization-delocalization transition of RBSOs as $\lambda$ varies. Our proof is based on a nontrivial extension of the $T$-expansion method and the concept of self-energy renormalization developed for random band matrices in arXiv:2104.12048. Additionally, we explore a conceptually new idea called coupling renormalization. This phenomenon extends the self-energy renormalization and is prevalent in quantum field theory, yet it is identified here for the first time within the context of random Schr\"odinger operators. We anticipate that our techniques can be adapted to real or non-Gaussian block potentials and more general interactions $\Psi$.