Higher Order Topological Systems: A New Paradigm

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.