High Spin-Chern-Number Insulator in $\alpha$-Antimonene with a Hidden Topological Phase

In investigating the topological electronic structures of monolayer $\alpha$-phase group V elements, we uncover a new topological phase, which is invisible in the symmetry-based topological quantum chemistry (TQC) as well as symmetry indicators (SIs). Since $\alpha$ phase As and Sb share the same band representations at high-symmetry points, they are both trivial insulators in terms of TQC and SIs. We demonstrate, however, that there is a topological phase transition between As and Sb that involves a band-gap closing at two $k$-points on the high-symmetry $\rm{X}$-$\rm{\Gamma}$-$\rm{X}$ line. In the absence of spin-orbit coupling (SOC), As is a trivial insulator, while Sb is a Dirac semimetal with four Dirac points (DPs) located away from the high-symmetry lines. Inclusion of $S_z$-conserved SOC gaps out the Dirac points and induces a nontrivial Berry curvature and drives Sb into a high spin Chern number topological phase. The band structure of $\alpha$-Bi differs from that of Sb by a band inversion at $\Gamma$, transforming Bi into a Z$_2$ topological insulator. Our study shows that quantized spin Hall conductivity can serve as a topological invariant beyond Z$_2$ for characterizing topological phases.