Constructing higher-order topological states in higher dimension

TL;DR

This paper demonstrates a method to construct higher-order topological states in higher dimensions by linking lower-dimensional edge states, using both theoretical models and experimental validation, and explores their relation to four-dimensional topological phases.

Contribution

It introduces a novel approach to connect lower and higher-order topological phases across dimensions through edge state manipulation and synthetic dimensions.

Findings

Successfully constructed 2D higher-order topological corner states from 1D edge states.

Connected 2D topological corner states to 4D topological phases via synthetic dimensions.

Provided experimental evidence supporting the theoretical model.

Abstract

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.