Equivalent topological invariants of topological insulators

A time-reversal (TR) invariant topological insulator can be generally defined by the effective topological field theory with a quantized θ coefficient, which can only take values of 0 or π. This theory is generally valid for an arbitrarily interacting system and the quantization of the θ invariant can be directly measured experimentally. Reduced to the case of a non-interacting system, the θ 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 TR invariant momenta. In this paper, we show the complete equivalence between the integral and the discrete invariants of the topological insulator.