Direct prediction of corner state configurations from edge winding numbers in 2D and 3D chiral-symmetric lattice systems

TL;DR

This paper introduces a method to predict corner state configurations in 2D and 3D chiral-symmetric systems using edge winding numbers, linking bulk topological invariants to boundary states.

Contribution

The work demonstrates how to directly predict corner state configurations from edge winding numbers in higher-order topological phases of 2D and 3D systems.

Findings

Winding numbers characterize corner states in 2D and 3D lattices.

Multiple topological phases coexist, including insulators and semimetals.

Experimental implementations are feasible with photonic lattices or electric circuits.

Abstract

Higher-order topological phases feature topologically protected boundary states in lower dimensions. Specifically, the zero-dimensional corner states are protected by the $d$th-order topology of a $d$-dimension system. In this work, we propose to predict different configurations of corner states from winding numbers defined for one-dimensional edges of the system. We first demonstrate the winding number characterization with a generalized two-dimensional square lattice belonging to the BDI symmetry class. In addition to the second-order topological insulating phase, the system may also be a nodal point semimetal or a weak topological insulator with topologically protected one-dimensional edge states coexisting with the corner states at zero energy. A three-dimensional cubic lattice with richer configurations of corner states is also studied. We further discuss several experimental implementations of our models with photonic lattices or electric circuits.