Anderson transitions

Abstract

The physics of Anderson transitions between localized and metallic phases in disordered systems is reviewed. The term "Anderson transition" is understood in a broad sense, including both metalinsulator transitions and quantum-Hall-type transitions between phases with localized states. The emphasis is put on recent developments, which include: multifractality of critical wave functions, criticality in the power-law random banded matrix model, symmetry classification of disordered electronic systems, mechanisms of criticality in quasi-one-dimensional and two-dimensional systems and survey of corresponding critical theories, network models, and random Dirac Hamiltonians. Analytical approaches are complemented by advanced numerical simulations.1 A remarkable exception is the behavior of ξ in disordered wires of the chiral symmetry, see Eqs. (5.21), (5.22).

Multifractal nature of the surface local density of states in three-dimensional topological insulators with magnetic and nonmagnetic disorder

“…[20][21][22] The multifractal spectrum τ (q) is determined by the most relevant (negative) 52 scaling dimension x q exhibited by an eigenoperator in this decomposition, and is given by 19,43 τ (q) = 2(q − 1) + x q − qx 1 .…”

“…24,29 This state exhibits strong multifractality that has been extensively studied in numerics. 18,19,[26][27][28] Fig. 3, but for different unitary class disorder strength combinations.…”

“…Termination can be viewed as a consequence of the restriction to positive sample measures f (α) ≥ 0. 19,38 In the localized regime, the states contributing to the LDOS at a given position in the sample have a discrete energy spectrum, quantized by the typical localization volume ξ 2 loc . As a result, all non-unity LDOS moments diverge in the absence of level smearing.…”

“…In QPI, the disorder is employed primarily as a facilitator to gleam information about the clean system. 11 Multifractal analysis [17][18][19] provides a complementary method better suited to extracting large-distance, disorder-dominated features in the same LDOS data field. It is a standard tool for assaying quantum interference phenomena, and is employed in the analysis of wavefunctions near a metal-insulator transition [18][19][20][21] as well as In the time-reversal invariant case, the impurities are neutral adatoms or charged dopant ions, depicted as spheres in (a).…”

Deconfinement, chiral transition and localisation in a QCD-like model

“…The curve λ c (T ) reaches zero at a temperature compatible with T c [7], as determined from thermodynamic observables [2,3]. The transition in the spectrum from localised to delocalised modes, taking place at the critical point λ c , was shown to be a genuine second-order phase transition [9], analogous to the metal-insulator transition in the Anderson model [38][39][40], which describes non-interacting electrons in a disordered crystal. Furthermore, in ref.…”