  Higher-order topological phase as a generalization of Berry phase attracts an
enormous amount of research. The current theoretical models supporting
higher-order topological phases, however, cannot give the connection between
lower 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 one
dimensional lattice. The two-dimensional square lattice owns independent
spatial modulation of coupling in each direction, and the combination of edge
states in each direction come up to the higher-order topological corner state
in two-dimensional lattice, revealing the connection of topological phase in
lower and higher dimensional lattices. Moreover, the topological corner states
in two-dimensional lattice can also be viewed as the dimension-reduction from a
four-dimensional topological phase characterized by vector Chern number,
considering two modulation phases as synthetic dimensions in Aubry-Andre-Harper
model discussed as example here. Our work deeps the understanding to
topological phases breaking through the lattice dimension, and provides a
promising tool constructing higher topological phases in higher dimensional
structures.
