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

Abstract

We generalize universal relations between the multifractal exponent α_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 Λ_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 Λ_c^o is universally related to the boundary multifractal exponent α_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.

13 pages, 7 figures

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• The general approach to the critical phase with coupled quasiperiodic chains
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• Numerical evidence for marginal scaling at the integer quantum Hall transition
• Quantum Hall transitions: An exact theory based on conformal restriction
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• Characterization of the Quantized Hall Insulator Phase in the Quantum Critical Regime
• Metal-insulator transition in a 2D system of chiral unitary class
• Exact exponents for the spin quantum Hall transition in the presence of multiple edge channels
• The scaling behavior of the insulator to plateau transition in topological band model
• Generalized multifractality at spin quantum Hall transition
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• Numerical Study of a Dual Representation of the Integer Quantum Hall Transition
• Conformal Invariance and Multifractality at Anderson Transitions in Arbitrary Dimensions
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• Generalized surface multifractality in 2D disordered systems
• Quantum multifractality as a probe of phase space in the Dicke model
• Mapping the Current-Current Correlation Function Near a Quantum Critical Point
• Many-body localization in Landau level subbands
• Critical exponent for the quantum spin Hall transition in Z_2 network model
• Composite Fermion Nonlinear Sigma Models
• Critical behavior at the integer quantum Hall transition in a network model on the Kagome lattice
• The integer quantum Hall transition: an $S$-matrix approach to random networks
• Multifractality in monitored single-particle dynamics
• Relevant perturbations at the spin quantum Hall transition
• Quantum Hall criticality in an amorphous Chern insulator
• Criticality without self-similarity: a 2D system with random long-range hopping
• Localization and interactions in topological and non-topological bands in two dimensions
• Localization and spin transport in honeycomb structures with spin-orbit coupling
• Elliptic mirror of the quantum Hall effect
• Edge state critical behavior of the integer quantum Hall transition
• Spin-mixing-tunneling network model for Anderson transitions in two-dimensional disordered spinful electrons
• Extreme-Value Criticality and Gain Decomposition at the Integer Quantum Hall Transition