Direct prediction of corner state configurations from edge winding numbers in 2D and 3D chiral-symmetric lattice systems

Higher-order topological phases feature topologically protected boundary states in lower dimensions. Specifically, the zero-dimensional corner states are protected by the $d$th-order topology of a $d$-dimension system. In this work, we propose to predict different configurations of corner states from winding numbers defined for one-dimensional edges of the system. We first demonstrate the winding number characterization with a generalized two-dimensional square lattice belonging to the BDI symmetry class. In addition to the second-order topological insulating phase, the system may also be a nodal point semimetal or a weak topological insulator with topologically protected one-dimensional edge states coexisting with the corner states at zero energy. A three-dimensional cubic lattice with richer configurations of corner states is also studied. We further discuss several experimental implementations of our models with photonic lattices or electric circuits.