Generalized Hall currents in topological insulators and superconductors

TL;DR

This paper extends the concept of quantized Hall currents to a broader class of topological materials, linking edge states to topological invariants and providing a calculational framework applicable across different dimensions and symmetries.

Contribution

It introduces a generalized current related to edge states in topological insulators and superconductors, applicable to various symmetry classes and dimensions, with potential extension to interacting theories.

Findings

Current quantization governed by topology in phase space

Applicable to ${f Z}$ and ${f Z}_2$ topological classes

Explicit examples with free relativistic fermions

Abstract

We generalize the idea of the quantized Hall current to count gapless edge states in topological materials, applying equally well to theories in different dimensions, with or without continuous symmetries in the bulk or chiral anomalies on the boundaries. This current is related to the index of the Euclidean fermion operator and can be calculated via one-loop Feynman diagrams. Quantization of the current is shown to be governed by topology in phase space, and the procedure can be applied to topological classes governed by either ${\mathbb Z}$ or ${\mathbb Z}_2$ invariants. We analyze several explicit examples of free fermions in relativistic field theories. We speculate that it may be possible to extend the technique to interacting theories as well, such as the interesting cases where interactions gap the edge states.