Topological Insulator and the $\theta$-vacuum in a system without boundaries

TL;DR

This paper investigates the effective action of boundary-less topological insulators, confirming the generation of a $ heta$-term at $ heta=\pi$ and its potential bulk effects, which diminish with infinite system size.

Contribution

It demonstrates the generation of a $ heta$-term in boundary-less topological insulators and analyzes its observable consequences in the bulk.

Findings

A uniform $ heta$-term with $ heta=\pi$ is generated in such systems.

The $ heta$-term's observable effects in the bulk vanish as system size approaches infinity.

The results apply to both one and three-dimensional topological insulators.

Abstract

In this paper we address two questions concerning the effective action of a topological insulator in one and three dimensional space without boundaries, such as a torus. The first is whether a uniform $\theta$-term with $\theta=\pi$ is generated for a strong topological insulator. The second is whether such a term has observable consequences in the bulk. The answers to both questions are positive, but the observability in three dimension vanishes for infinite system size.