Equivalent topological invariants of topological insulators

TL;DR

This paper demonstrates the complete equivalence between integral and discrete topological invariants in time-reversal invariant topological insulators, unifying different classification methods for these materials.

Contribution

It establishes the full equivalence between integral and discrete topological invariants, bridging different approaches to classify topological insulators.

Findings

Integral and discrete invariants are fully equivalent.

The topological field theory with quantized heta applies broadly.

Experimental measurement of the heta invariant is feasible.

Abstract

A time-reversal invariant topological insulator can be generally defined by the effective topological field theory with a quantized \theta coefficient, which can only take values of 0 or \pi. This theory is generally valid for an arbitrarily interacting system and the quantization of the \theta invariant can be directly measured experimentally. Reduced to the case of a non-interacting system, the \theta invariant can be expressed as an integral over the entire three dimensional Brillouin zone. Alternatively, non-interacting insulators can be classified by topological invariants defined over discrete time-reversal invariant momenta. In this paper, we show the complete equivalence between the integral and the discrete invariants of the topological insulator.