Higher-Order Topological Phases Hidden in the 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 Z₂ topological insulators. This commonly held viewpoint, however, hides the far richer nature of the QSHI state. Unlike the Z₂ 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 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 monolayer α-Sb as an example of a real material, which is time-reversal symmetric, and show that it carries an even spin Chern number C_s = 2. This unique phase of α-Sb 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. Our study unveiled the presence of higher-order topological phases that have so far remained hidden in the QSHI phase. Furthermore, our work suggests that the high spin Chern number insulators provide a new basis for developing materials platforms for efficient spintronic devices.