Effective field theories for topological insulators by functional bosonization

TL;DR

This paper develops effective field theories for topological insulators using functional bosonization, covering both non-interacting and fractional phases across various dimensions and symmetry classes.

Contribution

It derives BF-type topological field theories for non-interacting topological insulators and extends the framework to fractional phases via parton constructions.

Findings

Derived BF-type topological field theories for complex symmetry classes.

Obtained topological theories for $ ext{Z}_2$ invariants through dimensional reduction.

Discussed potential effective theories for fractional topological insulators.

Abstract

Effective field theories that describes the dynamics of a conserved U(1) current in terms of "hydrodynamic" degrees of freedom of topological phases in condensed matter are discussed in general dimension $D=d+1$ using the functional bosonization technique. For non-interacting topological insulators (superconductors) with a conserved U(1) charge and characterized by an integer topological invariant [more specifically, they are topological insulators in the complex symmetry classes (class A and AIII) and in the "primary series" of topological insulators in the eight real symmetry classes], we derive the BF-type topological field theories supplemented with the Chern-Simons (when $D$ is odd) or the $\theta$-term (when $D$ is even). For topological insulators characterized by a $\mathbb{Z}_2$ topological invariant (the first and second descendants of the primary series), their topological field theories are obtained by dimensional reduction. Building on this effective field theory description for non-interacting topological phases, we also discuss, following the spirit of the parton construction of the fractional quantum Hall effect by Block and Wen, the putative "fractional" topological insulators and their possible effective field theories, and use them to determine the physical properties of these non-trivial quantum phases.