Domain Walls in Topological Tri-hinge Matter

TL;DR

This paper explores the classification and engineering of domain walls in topological tri-hinge systems using graph theory and topological invariants, proposing models and material candidates for robust gapless states.

Contribution

It introduces a classification scheme for tri-hinge topological systems based on Euler characteristic and constructs a symmetry-invariant Hamiltonian model.

Findings

Classified skeleton matrices by Euler characteristic into three topological sets.

Built a tri-hinge Hamiltonian model with specific symmetry invariances.

Suggested candidate materials for realizing tri-hinge topological states.

Abstract

Using a link between graph theory and the geometry hosting higher order topological matter, we fill part of the missing results in the engineering of domain walls supporting gapless states for systems with three vertical hinges. The skeleton matrices which house the particle states responsible for the physical properties are classified by the Euler characteristic into three sets with topological index $\chi =0,1,2.$ A tri-hinge hamiltonian model invariant under the composite $\boldsymbol{M}_{1}\boldsymbol{T}$, $\boldsymbol{M}_{2}\boldsymbol{T}$, $\boldsymbol{M}_{3}\boldsymbol{T}$ is built. In this framework, $\boldsymbol{T}$ is the time reversing symmetry obeying $\boldsymbol{T}^{2}=-I$ and the $\boldsymbol{M}_{i}$'s are the generators of the three reflections of the dihedral $\mathbb{D}_{3}$ symmetry of triangle. To capture the tri-hinge states, candidate materials are suggested, thus opening up a variety of possibilities for investigating and designing robust materials against disorder and deformation.