Inhomogeneous Fixed Point Ensembles Revisited

TL;DR

This paper revisits the classification of disordered systems' density of states near the mobility edge, comparing theoretical predictions with explicit results to understand inhomogeneous fixed point ensembles.

Contribution

It critically examines the 1976 scaling law prediction for inhomogeneous fixed point ensembles using recent explicit results.

Findings

The scaling law $ ext{μ}=d u-1$ is supported by explicit results.

Differences between homogeneous and inhomogeneous ensembles are clarified.

The behavior of the density of states varies across different symmetry classes.

Abstract

The density of states of disordered systems in the Wigner-Dyson classes approaches some finite non-zero value at the mobility edge, whereas the density of states in systems of the chiral and Bogolubov-de Gennes classes shows a divergent or vanishing behavior in the band centre. Such types of behavior were classified as homogeneous and inhomogeneous fixed point ensembles within a real-space renormalization group approach. For the latter ensembles the scaling law $\mu=d\nu-1$ was derived for the power laws of the density of states $\rho\propto|E|^\mu$ and of the localization length $\xi\propto|E|^{-\nu}$. This prediction from 1976 is checked against explicit results obtained meanwhile.