Edge-corner correspondence: Boundary-obstructed topological phases with chiral symmetry

The bulk-edge correspondence characterizes topological insulators and superconductors. We generalize this concept to the bulk-corner correspondence and the edge-corner correspondence in two dimensions. In the bulk-corner (edge-corner) correspondence, the topological number is defined for the bulk (edge), while the topological phase is evidenced by the emergence of zero-energy corner states. It is shown that the boundary-obstructed topological phases recently proposed are the edge-corner-correspondence type, while the higher-order topological phases are classified into the bulk-corner-correspondence type and the edge-corner-correspondence type. We construct a simple two-dimensional model exhibiting the edge-corner correspondence based on two Chern insulators having the $s$-wave, $d$-wave, and ${s}_{\ifmmode\pm\else\textpm\fi{}}$-wave pairings. Here, the emergence of zero-energy corner states in the topological phase is a result of the topological numbers defined for the edge Hamiltonians and not for the bulk Hamiltonian. It is pointed out that the emergence of zero-energy corner states is observable by measuring the impedance resonance in topological electric circuits.