Higher-order topological insulators and semimetals in generalized Aubry-Andr\'e-Harper models

Higher-order topological phases of matter have been extensively studied in various areas of physics. While the Aubry-Andr\'e-Harper model provides a paradigmatic example to study topological phases, it has not been explored whether a generalized Aubry-Andr\'e-Harper model can exhibit a higher-order topological phenomenon. Here, we construct a two-dimensional higher-order topological insulator with chiral symmetry based on the Aubry-Andr\'e-Harper model. We find the coexistence of zero-energy and nonzero energy corner-localized modes. The former is protected by the quantized quadrupole moment, while the latter by the first Chern number of the Wannier band. The nonzero-energy mode can also be viewed as the consequence of a Chern insulator localized on a surface. More interestingly, the non-zero energy corner mode can lie in the continuum of extended bulk states and form a bound state in the continuum of higher-order topological systems. We finally propose an experimental scheme to realize our model in electric circuits. Our study opens a door to further study higher-order topological phases based on the Aubry-Andr\'e-Harper model.