Conformal invariance, multifractality, and finite-size scaling at Anderson localization transitions in two dimensions

TL;DR

This paper establishes universal relations between multifractal exponents and finite-size scaling amplitudes at two-dimensional Anderson localization transitions, verified across various symmetry classes and boundary conditions.

Contribution

It generalizes universal relations between multifractal exponents and critical finite-size scaling amplitudes, including cases with vanishing density of states, and verifies these relations numerically.

Findings

Universal relation between boundary multifractal exponent and FSS amplitude.

Generalization of relations to symmetry classes with zero density of states.

Numerical verification across four different Anderson transition types.

Abstract

We generalize universal relations between the multifractal exponent \alpha_0 for the scaling of the typical wave function magnitude at a (Anderson) localization-delocalization transition in two dimensions and the corresponding critical finite size scaling (FSS) amplitude \Lambda_c of the typical localization length in quasi-one-dimensional (Q1D) geometry: (i) When open boundary conditions are imposed in the transverse direction of Q1D samples (strip geometry), we show that the corresponding critical FSS amplitude \Lambda_c^o is universally related to the boundary multifractal exponent \alpha_0^s for the typical wave function amplitude along a straight boundary (surface). (ii) We further propose a generalization of these universal relations to those symmetry classes whose density of states vanishes at the transition. (iii) We verify our generalized relations [Eqs. (6) and (7)] numerically for the following four types of two-dimensional Anderson transitions: (a) the metal-to-(ordinary insulator) transition in the spin-orbit (symplectic) symmetry class, (b) the metal-to-(Z_2 topological insulator) transition which is also in the spin-orbit (symplectic) class, (c) the integer quantum Hall plateau transition, and (d) the spin quantum Hall plateau transition.