Higher-order topological phases hidden in quantum spin Hall insulators

Topological materials burgeoned with the discovery of the quantum spin Hall insulators (QSHIs). Since their discovery, QSHIs have been viewed as being ${\mathbb{Z}}_{2}$ topological insulators. This commonly held viewpoint, however, hides the far richer nature of the QSHI state. Unlike the ${\mathbb{Z}}_{2}$ topological insulator, which hosts gapless boundary states protected by the time-reversal symmetry, the QSHI does not support gapless edge states because the spin-rotation symmetry breaks down in real systems. Here, we demonstrate that QSHIs hide higher-order topological insulator phases through two exemplar systems. We first consider the Kane-Mele model under an external field and show that it carries an odd spin Chern number ${\mathcal{C}}_{s}=1$. The model is found to host gapless edge states in the absence of Rashba spin-orbit coupling (SOC). But, a gap opens up in the edge spectrum when SOC is included, and the system turns into a higher-order topological insulator with in-gap corner states emerging in the spectrum of a nanodisk. We also discuss a time-reversal symmetric tight-binding model on a square lattice, and show that it carries an even spin Chern number ${\mathcal{C}}_{s}=2$. This unique phase has been taken to be topologically trivial because of its gapped edge spectrum. We show it supports in-gap corner states and hosts a higher-order topological phase.