Delocalization of a general class of random block Schr\"odinger operators

TL;DR

This paper proves delocalization and quantum ergodicity for a class of random block Schrödinger operators on high-dimensional tori, demonstrating an Anderson transition as the coupling parameter varies, using advanced renormalization techniques.

Contribution

It introduces a novel coupling renormalization method for analyzing delocalization in random block Schrödinger operators, extending previous self-energy renormalization approaches.

Findings

Proves delocalization and quantum ergodicity for RBSOs in high dimensions.

Establishes an Anderson transition as the coupling parameter varies.

Develops a new coupling renormalization technique inspired by quantum field theory.

Abstract

We consider a natural class of extensions of the Anderson model on $\mathbb Z^d$, called random block Schr\"odinger operators (RBSOs), defined on the $d$-dimensional torus $(\mathbb Z/L\mathbb Z)^d$. These operators take the form $H=V+\lambda\Psi$, where $V$ is a diagonal block matrix whose diagonal blocks are i.i.d. $W^d\times W^d$ GUE, representing a random block potential, $\Psi$ describes interactions between neighboring blocks, and $\lambda\ll 1$ is a small coupling parameter (making $H$ a perturbation of $V$). We focus on three specific RBSOs: (1) the block Anderson model, where $\Psi$ is the discrete Laplacian on $(\mathbb Z/L\mathbb Z)^d$; (2) the Anderson orbital model, where $\Psi$ is a block Laplacian operator; (3) the Wegner orbital model, where the nearest-neighbor blocks of $\Psi$ are themselves random matrices. Assuming $d\ge 7$ and $W\ge L^\varepsilon$ for a small constant $\varepsilon>0$, and under a certain lower bound on $\lambda$, we establish delocalization and quantum unique ergodicity for bulk eigenvectors, along with quantum diffusion estimates for the Green's function. Combined with the localization results of arXiv:1608.02922, our results rigorously demonstrate the existence of an Anderson localization-delocalization transition for RBSOs as $\lambda$ varies. Our proof is based on the $T$-expansion method and the concept of self-energy renormalization, originally developed in the study of random band matrices in arXiv:2104.12048. In addition, we introduce a conceptually novel idea, called coupling renormalization, which extends the notion of self-energy renormalization. While this phenomenon is well-known in quantum field theory, it is identified here for the first time in the context of random Schr\"odinger operators. We expect that our methods can be extended to models with real or non-Gaussian block potentials, as well as more general forms of interactions.