Z2 topological term, the global anomaly, and the two-dimensional symplectic symmetry class of Anderson localization

TL;DR

This paper investigates a $Z_2$ topological term in a 2D Dirac Hamiltonian with disorder, revealing its role in Anderson localization and potential implications for graphene and topological insulators.

Contribution

It identifies and computes the $Z_2$ topological term and global anomaly in the symplectic class, linking topological effects to Anderson localization phenomena.

Findings

Presence of a $Z_2$ topological term in the sigma model.

Numerical computation of the global anomaly via spectral flow.

Relevance to graphene and 2D boundaries of 3D topological insulators.

Abstract

We discuss, for a two-dimensional Dirac Hamiltonian with random scalar potential, the presence of a $Z_2$ topological term in the non-linear sigma model encoding the physics of Anderson localization in the symplectic symmetry class. The $Z_2$ topological term realizes the sign of the Pfaffian of a family of Dirac operators. We compute the corresponding global anomaly, i.e., the change in the sign of the Pfaffian by studying a spectral flow numerically. This $Z_2$ topological effect can be relevant to graphene when the impurity potential is long-ranged and, also, to the two-dimensional boundaries of a three-dimensional lattice model of $Z_2$ topological insulators in the symplectic symmetry class.