Edge states, corner states, and flat bands in a two-dimensional $\cal PT$ symmetric system

We study a two dimensional model including Su-Schrieffer-Heeger (SSH) type bond alternation as well as next-nearest neighbor hoppings yielding flat bands. The key symmetry of the model is space-time inversion ($\cal PT$) symmetry, which guarantees quantized Berry phases. This implies that edge states as well as corner states would show up if boundaries are introduced to the system. We also argue that an infinitesimal $\cal PT$ symmetry-breaking perturbation could drive flat bands into flat Chern bands.