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

Understand

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.