On the localization transition in three dimensions: Monte-Carlo simulation of a non-linear $\sigma$-model

TL;DR

This paper investigates the critical behavior of a supersymmetric non-linear sigma model related to the localization transition in disordered systems, providing numerical and analytical insights into phase transition characteristics in three dimensions.

Contribution

It combines analytical and Monte Carlo simulations to analyze the localization transition, offering new estimates of critical exponents and multifractal properties.

Findings

Localization length exponent determined

Inverse participation numbers calculated

Continuous phase transition confirmed

Abstract

We present a combination of analytical and numerical calculations for the critical behavior of a supersymmetric non-linear $\sigma$-model within the context of the localization transition of a disordered one-electron system. As a result, we obtain a localization length exponent and a set of inverse participation numbers in three dimensions. We find a continuous phase transition with the features of one-parameter scaling and multifractality at the critical point.