Renormalization group for Anderson localization on high-dimensional lattices

TL;DR

This paper explores how the critical properties of the Anderson localization transition depend on spatial dimension using renormalization group techniques, connecting different dimensional regimes and providing a framework for future disordered system studies.

Contribution

It introduces a unified renormalization group framework that reconciles $d=2$ and high-dimensional limits for Anderson localization, including the derivation of the irrelevant exponent from nonlinear sigma-model expansion.

Findings

The $eta$-function evolves smoothly from $d=2$ to high dimensions.

The $eta$-function form at the transition point is connected to the fractal dimension.

A conjecture is proposed for a lower bound of the fractal dimension.

Abstract

We discuss the dependence of the critical properties of the Anderson model on the dimension $d$ in the language of $\beta$-function and renormalization group recently introduced in Ref.[arXiv:2306.14965] in the context of Anderson transition on random regular graphs. We show how in the delocalized region, including the transition point, the one-parameter scaling part of the $\beta$-function for the fractal dimension $D_{1}$ evolves smoothly from its $d=2$ form, in which $\beta_2\leq 0$, to its $\beta_\infty\geq 0$ form, which is represented by the regular random graph (RRG) result. We show how the $\epsilon=d-2$ expansion and the $1/d$ expansion around the RRG result can be reconciled and how the initial part of a renormalization group trajectory governed by the irrelevant exponent $y$ depends on dimensionality. We also show how the irrelevant exponent emerges out of the high-gradient terms of expansion in the nonlinear sigma-model and put forward a conjecture about a lower bound for the fractal dimension. The framework introduced here may serve as a basis for investigations of disordered many-body systems and of more general non-equilibrium quantum systems.