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

## TL;DR

This paper investigates the emergence of corner and edge states in a two-dimensional $ ext{PT}$ symmetric lattice model with flat bands, highlighting the role of symmetry and perturbations in topological phenomena.

## Contribution

It introduces a 2D model with bond alternation and next-nearest neighbor hoppings demonstrating how $ ext{PT}$ symmetry leads to quantized Berry phases and topological states.

## Key findings

- Corner states appear on flat bands in the model.
- $ ext{PT}$ symmetry guarantees quantized Berry phases and topological edge/corner states.
- Infinitesimal $ ext{PT}$ symmetry-breaking can transform flat bands into flat Chern bands.

## Abstract

We study corner states on a flat band in the square lattice. To this end, we introduce a two dimensional model including Su-Schrieffer-Heeger type bond alternation responsible for corner states as well as next-nearest neighbor hoppings yielding flat bands. The key symmetry of the model for corner states 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.