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

Abstract

We generalize universal relations between the multifractal exponent ␣ 0 for the scaling of the typical wavefunction magnitude at a ͑Anderson͒ localization-delocalization transition in two dimensions and the corresponding critical finite-size-scaling ͑FSS͒ amplitude ⌳ c of the typical localization length in quasi-onedimensional ͑Q1D͒ geometry: ͑i͒ when open boundary conditions are imposed in the transverse direction of Q1D samples ͑strip geometry͒, we show that the corresponding critical FSS amplitude ⌳ c o is universally related to the boundary multifractal exponent ␣ 0 s for the typical wave-function amplitude along a straight boundary ͑surface͒. ͑ii͒ We further propose a generalization of these universal relations to those symmetry classes whose density of states vanishes at the transition. ͑iii͒ We verify our generalized relations ͓Eqs. ͑6͒ and ͑7͔͒ numerically for the following four types of two-dimensional Anderson transitions: ͑a͒ the metal-to-͑ordinary insulator͒ transition in the spin-orbit ͑symplectic͒ symmetry class, ͑b͒ the metal-to-͑Z 2 topological insulator͒ transition which is also in the spin-orbit ͑symplectic͒ class, ͑c͒ the integer quantum-Hall plateau transition, and ͑d͒ the spin quantum-Hall plateau transition.

Geometrically disordered network models, quenched quantum gravity, and critical behavior at quantum Hall plateau transitions

“…40 The regular geometry of the CC model allows one to apply numerical transfer matrix techniques. 41,42 Recent implementations of this [43][44][45][46][47][48] and other methods 49,50 agree on the value ν in the range 2.56-2.62, certainly different from ν exp . The discrepancy points to the importance of the long-range electron-electron interaction, which certainly affects the scaling near the integer QH transition [51][52][53][54][55][56][57][58] and is relevant for the interpretation of experiments.…”

Critical exponent for the Anderson transition in the three-dimensional orthogonal universality class

“…The critical exponent ν has been well studied both experimentally [33] and numerically. For the Chalker-Coddington model [34,35], the exponent is estimated to be ν ≈ 2.6 [36][37][38][39][40] (and see table 6). The universality of this value is supported by a study of the quantum Hall transition using a periodically driven Hamiltonian model [41].…”

Multifractality and electron-electron interaction at Anderson transitions

“…On the theoretical side, our paper paves a way to a systematic investigation of multifractality at interacting critical points of localization transitions within the nonlinear sigma model approach. The rich physics related to multifractality in the absence of interaction, including, in particular, systems of different symmetry classes and different dimensionalities, symmetries of mutlifractal spectra, termination and freezing, implications of conformal symmetry, connection to entanglement entropy, and manifestation of multifractality in various observables [2,7,[45][46][47] remains to be explored in the presence of Coulomb interaction. Finally, we mention that our analysis of multifractal correlations in the local density of states can be extended to superconductor-insulator transitions [48].…”