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

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

2009-11-29

arxiv.org/abs/0911.5443v3

Phys. Rev. B 82, 035309 (2010)

Physics

Condensed Matter

Disordered Systems and Neural Networks

13 pages, 7 figures

Scientific paper

10.1103/PhysRevB.82.035309

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.

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