Higher Order Topological Systems: A New Paradigm

TL;DR

Higher order topological insulators are a novel class of materials with boundary modes of lower dimension than traditional topological insulators, protected by crystalline symmetries, and can host Majorana modes in superconducting states.

Contribution

This paper introduces the concept of higher order topological insulators and superconductors, detailing their boundary modes, symmetry protections, and experimental realizations, expanding the topological phases framework.

Findings

Higher order topological insulators host boundary modes of dimension (m - n).

Experimental observations include acoustic systems and layered materials like WTe2.

Presence of Majorana modes in higher order topological superconductors.

Abstract

Higher order topological insulators are a new class of topological insulators in dimensions $\rm d>1$. These higher-order topological insulators possess $\rm (d - 1)$-dimensional boundaries that, unlike those of conventional topological insulators, do not conduct via gapless states but instead are themselves topological insulators. Precisely, an $\rm n^{\rm th}$-order topological insulator in $\rm m$ dimensions hosts $\rm d_{c} = (m - n)$-dimensional boundary modes $\rm (n \leq m)$. For instance, a three-dimensional second (third) order topological insulator hosts gapless modes on the hinges (corners), characterized by $\rm d_{c} = 1 (0)$. Similarly, a second order topological insulator in two dimensions only has gapless corner states ($\rm d_{c} = 0$) localized at the boundary. These higher order phases are protected by various crystalline symmetries. Moreover, in presence of proximity induced superconductivity and appropriate symmetry breaking perturbations, the above mentioned bulk-boundary correspondence can be extended to higher order topological superconductors hosting Majorana hinge or corner modes. Such higher-order systems constitute a distinctive new family of topological phases of matter which has been experimentally observed in acoustic systems, multilayer $\rm WTe_{2}$ and $\rm Bi_{4}Br_{4}$ chains. In this general article, the basic phenomenology of higher order topological insulators and higher order topological superconductors are presented along with some of their experimental realization.