Quantized index for topological phase transitions in a generic finite momentum region

When a topological phase transition occurs between distinct insulator phases, a gap-closing always occurs between the valence and conduction bands. In general, wave functions change only around the gap-closing momentum when comparing just before and just after the gap-closing. This fact indicates that examining the wave functions around the gap-closing momentum is enough to know whether a topological phase transition occurs, even though the topological invariant would be defined by an integral in the whole momentum space. In fact, in several cases with high crystalline symmetries, a topological phase transition can be detected by checking irreducible representations of wave functions at high symmetry points, where a gap-closing can occur in the system symmetry.

In this presentation, we introduce a quantized index by examining the wave functions in a finite momentum region for general cases without crystalline symmetries, and show it works as a detector of a topological phase transition. In cases without crystalline symmetries, the gap-closing occurs in a generic momentum and thus the irreducible representation cannot be used as a detector of the topological phase transition. We have applied the recently proposed method of using the Gauss-Bonnet theorem on a manifold with a boundary, and have obtained a quantized index from a generic and finite momentum region. We also have shown that the obtained index corresponds to the difference of topological invariant between before and after the gap-closing and thus it can be used as a detector of a topological phase transition. Further, we also find that, when we calculate an invariant of topological crystalline insulators, we can skip the evaluation of symmetry eigenvalues to make the calculation scheme easy.