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

#### 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 $\Lambda_c0$ is universally related to the boundary multifractal exponent $\alpha_0s$ 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.