Delocalization of a general class of random block Schrödinger operators

TL;DR

The paper establishes an Anderson-like localization-delocalization framework for a broad class of random block Schrödinger operators (RBSOs) on high-dimensional tori by introducing a block-structured perturbative scheme. Central to the analysis is the $T$-expansion and a novel coupling renormalization mechanism that enables sharp local laws, quantum unique ergodicity, and quantum diffusion for bulk states in dimensions $d\ge 7$ under weak coupling ($\lambda$ small) and sufficiently large block size $W$. The authors prove an explicit delocalization (with QUE) and diffusive transport up to the Thouless time, and connect these results with earlier localization findings to rigorously demonstrate the existence of an Anderson localization-delocalization transition as $\lambda$ varies. Methodologically, the work introduces a hierarchy of graph expansions, self-energy renormalization, and a vertex-cancellation phenomenon that extends techniques from random band matrices to more general, non-translation-invariant block models. The framework promises robustness to non-Gaussian block potentials and more general inter-block interactions, offering a principled route to understanding transport in highly structured disordered systems.

Abstract

We consider a natural class of extensions of the Anderson model on $\mathbb Z^d$, called random block Schrödinger operators (RBSOs), defined on the $d$-dimensional torus $(\mathbb Z/L\mathbb Z)^d$. These operators take the form $H=V+λΨ$, 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, $Ψ$ describes interactions between neighboring blocks, and $λ\ll 1$ is a small coupling parameter (making $H$ a perturbation of $V$). We focus on three specific RBSOs: (1) the block Anderson model, where $Ψ$ is the discrete Laplacian on $(\mathbb Z/L\mathbb Z)^d$; (2) the Anderson orbital model, where $Ψ$ is a block Laplacian operator; (3) the Wegner orbital model, where the nearest-neighbor blocks of $Ψ$ 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 $λ$, 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 $λ$ 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 Schrödinger 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.

Figures (2)

• Figure 1: We use undirected black solid edges to represent the blue and red solid edges in the graph. In the second graph, the edges $e_1$, $e_2,$$e_3,$$e_4$ are long solid edges of lengths $2,\ 3, \ 2$, $3$, respectively. In the second equality, we decompose the weight (i.e., self-loop) $e_4$ into a light weight (i.e., self-loop with a $\circ$) plus a scalar. In the latter case, we can further reduce $f_1$ and $f_2$ to a long solid edge $e_5$. The self-loop $e_5$ is then decomposed into a light weight and a scalar, which gives the second and third graphs in the third equality. In the last graph, we have also replaced $f_3$ and $f_4$ with another long solid edge $e_6$ of length 2.
• Figure 2: A tree-like molecular graph, where $x_i$'s denote external molecules and $a_i$'s denote internal molecules. The black double-line edges represent the diffusive edges in the black net, and the blue lines represent the diffusive or free edges in the blue net. Here, the dashed lines represent the redundant edges in the generalized doubly connected property.

Theorems & Definitions (167)

• Definition 1.1: Lattice of blocks
• Definition 1.2: Random block Schrödinger operators
• Remark 1.3
• Theorem 1.4: Localization
• proof
• Definition 1.5: Stochastic domination and high probability event
• Lemma 1.6: Ward's identity
• Definition 2.1: Matrix limit of $G$
• Theorem 2.2: Local law
• Corollary 2.3: Weak delocalization of bulk eigenvectors