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

Obuse H, Subramaniam AR, Furusaki A, Gruzberg IA, Ludwig AWW. Physical Review B, 82, 035309, 2010


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.