Finite-size scaling and multifractality at the Anderson transition for the three Wigner-Dyson symmetry classes in three dimensions

TL;DR

This study uses large-scale numerical simulations to analyze the multifractal properties and critical exponents of the Anderson transition across three symmetry classes in three dimensions, enhancing understanding of the transition's universality.

Contribution

It provides the first comprehensive numerical analysis of multifractal exponents for all three Wigner-Dyson symmetry classes in 3D, filling a gap in the literature.

Findings

Critical exponents agree with previous high-precision results.

Multifractal spectra differ among symmetry classes.

Finite-size scaling confirms universality of the Anderson transition.

Abstract

The disorder induced metal--insulator transition is investigated in a three-dimensional simple cubic lattice and compared for the presence and absence of time-reversal and spin-rotational symmetry, i.e. in the three conventional symmetry classes. Large scale numerical simulations have been performed on systems with linear sizes up to $L=100$ in order to obtain eigenstates at the band center, $E=0$. The multifractal dimensions, exponents $D_q$ and $\alpha_q$, have been determined in the range of $-1\leq q\leq 2$. The finite-size scaling of the generalized multifractal exponents provide the critical exponents for the different symmetry classes in accordance with values known from the literature based on high precision transfer matrix techniques. The multifractal exponents of the different symmetry classes provide further characterization of the Anderson transition, which was missing from the literature so far.