Constructing higher-order topological states in higher dimensions

Higher-order topological phases as the generalization of Berry phases attract an enormous amount of research. The current theoretical models supporting higher-order topological phases, however, cannot give the connection between lower-order and higher-order topological phases when extending the lattice from lower to higher dimensions. Here, we theoretically propose and experimentally demonstrate a topological corner state constructed from the edge states in a one-dimensional lattice. The two-dimensional square lattice owns independent spatial modulation of coupling in each direction, and the combination of edge states in two directions comes up to the higher-order topological corner state in a two-dimensional lattice, revealing the connection of topological phase in lower- and higher-dimensional lattices. Our work deepens the understanding of the topological phases breaking through the lattice dimensions, and provides a promising tool constructing higher topological phases in higher-dimensional structures.