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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 model exhibiting the edge-corner correspondence based on two Chern insulators having the $s$-wave, $d$-wave and $s_{\pm }$-wave pairings. It is possible to define topological numbers for the edge Hamiltonians, and we have zero-energy corner states in the topological phase. The emergence of zero-energy corner states is observable by measuring the impedance resonance in topological electric circuits.