Disorder and Topology: Anderson Transitions

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According to traditional theory, the critical behavior at the Anderson transition (i.e. the quantum phase transition between localized and delocalized energy eigenstates of a disordered electron system) is captured by a one-parameter scaling hypothesis. First put forward in a celebrated paper by Abrahams et al., this hypothesis received brilliant confirmation from weak-coupling field-theory computations (carried out near the lower critical dimension of d=2) for the nonlinear sigma model of Wegner and Efetov. Now, while the Anderson transition is under good control in the weak-coupling regime just above two dimensions, there exists mounting analytical and numerical evidence that the traditional picture needs modification in high dimension and, more generally, in situations where the Anderson transition occurs at strong coupling (i.e. for a critical conductance of the order of the conductance quantum). The emerging modification is rather striking: going beyond the paradigm of one-parameter scaling, Altshuler, Kravtsov and collaborators (and independently, the present author) have argued that there should exist a third stable phase distinct from the two known phases (namely, the metal and the Anderson insulator). Moreover, in our recent work we have pointed out that the third phase is set apart by three characteristic features which are inter-related: (i) fractal energy eigenstates, (ii) singular continuous energy spectrum (of the disordered electron Hamiltonian in infinite volume), and (iii) a novel scenario of partial symmetry breaking (in the sense of the Landau theory of phases and phase transitions) in the field-theory formulation of the disordered electron system. The overarching objective of the present project is to develop a viable field-theory approach that is able to handle strong-coupling Anderson transitions (and related problems) in a controlled way. In the short term, our concrete goals are (i) to derive a two-parameter field theory (superseding the nonlinear sigma model) for Anderson transitions in high dimension, (ii) to develop the complete scaling theory (including relevant and irrelevant perturbations) of the integer quantum Hall transition, (iii) to formulate the conformal field theory describing the scaling limit of the spin quantum Hall transition (a.k.a. class C), and (iv) to explain the spectrum-wide quantum criticality of surface states, which has been predicted for topological insulators of symmetry class AIII. Our far goal is to develop the analytical theory to such a mature stage that the existing puzzles and controversies ("Do Anderson transitions violate the principle of conformal symmetry?", "What is the correct ansatz to use with two-parameter scaling in the case of tree-like graphs?", to name a few) can all be decided by further numerical work.