\(\pi\) quantization in the Berry phase by the multiple degenerate points

Semiconductors have a valence band consisting of multi-subbands. Inter-subband interaction causes multiple degenerate points near the Brillouin zone center \(\Gamma\) $\mathrm{\Gamma }$ even when the system exhibits space-reversal asymmetry. In the present study, we analyzed the effect of the existence of multiple degenerate points on topological features using Berry parameter investigation. For reasonable generalization, we propose a model Hamiltonian that represents multiple degeneracies in a two-dimensional quantum-well system. We also explored “topological” modifications through numerical calculations of the effective magnetic field and Berry’s connection and curvature. We found that the energy dependence of the Berry phase is affected by the interaction between degenerate points.