Systematic study for two-dimensional Z2 topological phase transitions at high-symmetry points in all layer groups

We construct a general theory of Z2 topological phase transitions in two-dimensional systems with time -reversal symmetry. We investigate the possibilities of Z2 topological phase transitions at band inversions at all high-symmetry points in k space in all 80 layer groups. We exclude the layer groups with inversion symmetry because the Z2 topological phase transition is known to be associated with band inversions with an exchange of parities. Among the other layer groups, we find 21 layer groups with insulator-to-insulator transitions with band inversion, and this problem is finally reduced to five point groups C3, C4, C6, S4, and C3h. We show how the change of the Z2 topological invariant at a band inversion is entirely determined by the irreducible representations of occupied and unoccupied bands at the high-symmetry point. For example, in the case of C3, we show that the Z2 topological invariants change whenever the band inversion occurs between two Kramers pairs whose C3 eigenvalues are {e pi i/3, e-pi i/3} and {-1, -1}. These results are not included in the theory of symmetry-based indicators or topological quantum chemistry.