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

Abstract

We generalize universal relations between the multifractal exponent ${\ensuremath{\alpha}}_{0}$ 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 ${\ensuremath{\Lambda}}_{c}$ 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 ${\ensuremath{\Lambda}}_{c}^{o}$ is universally related to the boundary multifractal exponent ${\ensuremath{\alpha}}_{0}^{s}$ 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-(${\mathbb{Z}}_{2}$ 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.