Critical regime of two dimensional Ando model: relation between critical conductance and fractal dimension of electronic eigenstates

TL;DR

This paper investigates the critical conductance and fractal properties of electronic eigenstates in a disordered two-dimensional spin-orbit coupled system, establishing a quantitative relation and estimating the critical exponent.

Contribution

It provides a numerical verification of the relation between critical conductance and fractal dimension, and estimates the critical exponent for the 2D Ando model.

Findings

Critical conductance g_c ≈ 1.42 e^2/h

Fractal dimension D(1) ≈ 1.889

Critical exponent ν ≈ 2.80

Abstract

The critical two-terminal conductance $g_c$ and the spatial fluctuations of critical eigenstates are investigated for a disordered two dimensional model of non-interacting electrons subject to spin-orbit scattering (Ando model). For square samples, we verify numerically the relation $\sigma_c=1/[2\pi(2-D(1))] e^2/h$ between critical conductivity $\sigma_c=g_c=(1.42\pm 0.005) e^2/h$ and the fractal information dimension of the electron wave function, $D(1)=1.889\pm 0.001$. Through a detailed numerical scaling analysis of the two-terminal conductance we also estimate the critical exponent $\nu=2.80\pm 0.04$ that governs the quantum phase transition.