Topology, Delocalization via Average Symmetry and the Symplectic Anderson Transition

TL;DR

This paper develops a field theory for the Anderson transition in 2D disordered systems with spin-orbit interactions, highlighting the role of topological defects and symmetry in localization and delocalization phenomena.

Contribution

It introduces a topological field theory framework that explains the Anderson transition via vortex defects and symmetry considerations, connecting to the Kosterlitz Thouless transition.

Findings

Critical conductivity and correlation length exponent computed in epsilon expansion.

Two distinct transitions between metallic and insulating phases identified.

Delocalized surface states explained by symmetry restoration after disorder averaging.

Abstract

A field theory of the Anderson transition in two dimensional disordered systems with spin-orbit interactions and time-reversal symmetry is developed, in which the proliferation of vortex-like topological defects is essential for localization. The sign of vortex fugacity determines the $Z_2$ topological class of the localized phase. There are two distinct, but equivalent transitions between the metallic phase and the two insulating phases. The critical conductivity and correlation length exponent of these transitions are computed in a $N=1-\epsilon$ expansion in the number of replicas, where for small $\epsilon$ the critical points are perturbatively connected to the Kosterlitz Thouless critical point. Delocalized states, which arise at the surface of weak topological insulators and topological crystalline insulators, occur because vortex proliferation is forbidden due to the presence of symmetries that are violated by disorder, but are restored by disorder averaging.