Universality classes of the Anderson transition in three-dimensional symmetry classes AIII, BDI, C, D and CI

TL;DR

This paper investigates the universal critical properties of Anderson transitions in five three-dimensional symmetry classes, determining their critical exponents through transfer matrix calculations and finite-size scaling.

Contribution

It provides the first comprehensive numerical determination of the critical exponents for Anderson transitions in five nonstandard 3D symmetry classes, including classes AIII, BDI, C, D, and CI.

Findings

Critical exponents for classes AIII, D, and C are precisely estimated.

Critical exponents for classes CI and BDI are obtained, with some results consistent with previous studies.

The study confirms universality of the Anderson transition in these symmetry classes.

Abstract

We clarify universal critical properties of delocalization-localization transitions in three-dimensional (3D) unitary and orthogonal classes with particle-hole and/or chiral symmetries (classes AIII, BDI, D, C and CI). We first introduce tight-binding models on cubic lattice that belong to these five nonstandard symmetry classes respectively. Unlike the Bogoliubov-de Gennes Hamiltonian for superconductors, all the five models have finite areas of Fermi surfaces in the momentum space in the clean limit. Thereby, the scaling theory of the Anderson transition guarantees the presence of the delocalization-localization transitions at finite disorder strength in these models. Based on this expectation, we carry out extensive transfer matrix calculations of the Lyapunov exponents for zero-energy eigenstates of the disordered tight-binding models with quasi-one-dimensional geometry. Near the Anderson transition point, the correlation length diverges with a universal critical exponent $\nu$. Using finite-size scaling analysis of the localization length, we determine the critical exponent of the transitions and scaling function of the (normalized) localization length in the three non-standard unitary symmetry classes: $\nu_{\text{AIII}}=1.06\pm0.02$, $\nu_{\text{D}}=0.87\pm0.03$, and $\nu_{\text{C}}=0.996\pm0.012$. Our result of the class C is consistent with a previous study of classical network model [M. Ortu\~no et al, Phys. Rev. Lett. 102, 070603 (2009)]. The critical exponents of the two non-standard orthogonal classes are estimated as $\nu_{\text{CI}}=1.17 \pm 0.02$ and $\nu_{\text{BDI}}=1.12 \pm 0.06$. Our result of the class CI is consistent with another previous study of a lattice model, while the exponent of the class BDI is at variance with the previous evaluation of nodal Dirac ring model [X. L. Luo et al, Phys. Rev. B. 101, 020202(R) (2020)].