  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.
