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

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 $θ$-term with $θ=π$ 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.

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