Scaling and renormalization in the modern theory of polarization: application to disordered systems

TL;DR

This paper develops a scaling and renormalization framework using polarization amplitude to analyze disordered systems, revealing fixed points and phase transitions in various dimensions.

Contribution

It introduces a novel scaling theory and renormalization method based on polarization amplitude, applied to disordered systems in one, two, and three dimensions.

Findings

In 1D, disorder tends to infinity, indicating absence of extended states.

In 3D, a metal-insulator transition occurs at finite disorder.

In 2D, fixed points shift with system size, suggesting size-dependent behavior.

Abstract

We develop a scaling theory and a renormalization technique in the context of the modern theory of polarization. The central idea is to use the characteristic function (also known as the polarization amplitude) in place of the free energy in the scaling theory and in place of the Boltzmann probability in a position-space renormalization scheme. We derive a scaling relation between critical exponents which we test in a variety of models in one and two dimensions. We then apply the renormalization to disordered systems. In one dimension the renormalized disorder strength tends to infinity indicating the entire absence of extended states. Zero(infinite) disorder is a repulsive(attractive) fixed point. In two and three dimensions, at small system sizes, two additional fixed points appear, both at finite disorder, $W_a$($W_r$) is attractive(repulsive) such that $W_a<W_r$. In three dimensions $W_a$ tends to zero, $W_r$ remains finite, indicating metal-insulator transition at finite disorder. In two dimensions we are limited by system size, but we find that both $W_a$ and $W_r$ decrease significantly as system size is increased.