Emergent spin Hall quantization and high-order van Hove singularities in square-octagonal MA$_2$Z$_4$

Abstract

Quantum spin Hall (QSH) insulators are versatile platforms for exploring exotic quantum phases, especially when combined with high-order van Hove singularities (VHSs) that enhance electron correlations. However, perfect spin Hall quantization is often hindered by spin mixing from strong spin-orbit coupling, and the emergence of such VHSs is highly sensitive to material-specific electronic structures. Here, we predict a class of seven-layered square-octagonal MA$_2$Z$_4$ (M = Mo/W, A = Si/Ge, Z = Pnictogen) isomers that host a robust, large-gap QSH phase with nearly quantized spin Hall conductivity and intrinsic high-order VHSs. Topological and symmetry analyses reveal that compounds with Z = P, As, and Sb are $\mathbb{Z}_2$ nontrivial with spin Chern number $C_s = 1$ and support $S_z$-polarized edge states, while those with Z = N are trivial insulators. The QSH phase features an $S_z$-conserving spin Hamiltonian consistent with an emergent spin $\mathrm{U}(1)$ quasi-symmetry, yielding spin Hall conductivity $\sim 2e^2/h$. Notably, MA$_2$(As, Sb)$_4$ compounds exhibit quasi-flat bands near the Fermi level in the inverted regime, with WSi$_2$Sb$_4$ additionally hosting four high-order VHSs at generic momentum points. These results position square-octagonal MA$_2$Z$_4$ materials as robust QSH insulators for realizing quantized spin Hall conductivity and correlated topological phases, including fractionalized states and possibly non-Abelian anyons.

Figures (10)

• Figure 1: Atomic structure and electronic features of MA$_2$Z$_4$. (a) Hexagonal 1H and (b) square-octagonal (SO) phases, with unit cells marked in black. (c) Side view of the SO phase showing Z–A–Z–M–Z–A–Z stacking, analogous to the 1H structure. Local coordination includes distorted MA$_6$ octahedra and AZ$_4$ tetrahedra. (d) In the 1T$^\prime$ phase, $d$–$p$ band inversion creates spinless Dirac cones with linear dispersion at the $\Lambda$ points along $k_y$. (e) In the SO phase, $d$-$d$ band inversion leads to a quadratic spinless Dirac cone that gaps under SOC, generating quasi-flat bands and van Hove singularities (VHSs) at $\delta_{vhs}$. (f–g) Comparison of spin U(1) symmetry, where the spin axis $\mathbf{S}$ aligns with the $z$-axis, and spin U(1) quasi-symmetry, where $\mathbf{S}$ tilts slightly away from $z$. This tilt leads to deviations from spin Hall quantization in device geometries misaligned with $\mathbf{S}$, and defines the optimal orientation for observing a quantized response.
• Figure 2: Band structure and topological characterization. Orbital-resolved band structure of (a) WSi$_2$N$_4$, (b) WSi$_2$P$_4$, and (c) WSi$_2$Sb$_4$ with spin--orbit coupling (SOC). Red and blue mark transition-metal $d_{x^2-y^2}$ and $d_{z^2}$ orbitals. Irreducible representations (IRs) and associated parities at the $\Gamma$ point are shown. (d)--(e) Closeup band structures of WSi$_2$As$_4$ and WSi$_2$Sb$_4$ without and with SOC. The doubly degenerate $\Gamma_5^-$ at the Fermi level splits upon inclusion of SOC, opening a gap and driving the system into a QSH phase with $\mathbb{Z}_2=1$. WSi$_2$Sb$_4$ exhibits flatter energy dispersion near the Fermi level with VHSs. (f) (010) edge spectrum of WSi$_2$As$_4$. (g) Schematic band evolution across the MA$_2$Z$_4$ compounds. Four bands near the Fermi level form a trivial ordering with $\mathbb{Z}_2 = 0$ for $Z=$ N. Substituting heavier pnictogens ($Z=$ P, As, Sb) induces a band inversion between the $\Gamma_3^+$ and $\Gamma_5^-$ states, forming spinless Dirac cones without SOC. Inclusion of SOC opens an inverted gap in $\Gamma_5^-$, yielding a QSH state with $\mathbb{Z}_2 = 1$.
• Figure 3: Feature spectrum topology and spin U(1) quasi-symmetry. (a) Spin feature spectrum ($P\hat{S}_z P$) of the occupied bulk states of MoSi$_2$As$_4$ and WSi$_2$As$_4$. Eigenvalues remain nearly flat and pinned at $\pm \hbar/2$, with slight deviations near the $\Gamma$ point. The zoomed view shows a larger deviation in WSi$_2$As$_4$, indicating stronger spin-axis tilt from the $z$ direction. The spin gap $\Delta_{S_z} \sim \hbar$ reflects an spin U(1) quasi-symmetry. (b) Spin-resolved Wilson loop spectrum of WSi$_2$As$_4$ showing Chern numbers $+1$ and $-1$ for spin-up and spin-down sectors, giving a total spin Chern number $C_S = 1$. (c) (010) edge state spin texture of WSi$_2$As$_4$ showing counter-propagating $S_z$-polarized edge states. (d) Intrinsic SHC $\sigma_{xy}^z$ of MoSi$_2$As$_4$ and WSi$_2$As$_4$ with a nearly quantized plateau of $\sim 2e^2/h$ within the bulk gap.
• Figure 4: High-order VHSs in WSi$_2$Sb$_4$. (a) Density of states (DOS) showing two sharp peaks associated with VHSs (VHS$_1$ and VHS$_2$) at $-19$ meV and $102$ meV near the Fermi level. (b) Valence band energy dispersion reveals quasi-flat regions with four symmetry-related saddle-point VHSs at generic $k$ points along the $\Gamma-S$ directions. The constant energy contour marks these VHSs locations. (c) Local dispersion near VHS$_1$ along principal axes $k_a$ and $k_b$, with polynomial fits: fourth-order along $k_a$ (blue) and second-order along $k_b$ (red), indicating higher-order VHSs. (d) (010) edge spectrum showing coexistence of VHSs and topological edge states.
• Figure S1: Thermodynamical stability of square-octagonal (SO) monolayers. Free energy as a function of simulation time during ab initio molecular dynamics simulations at $T=300$ K for monolayers (a) WSi$_2$P$_4$ and (b) WSi$_2$As$_4$. The red line represents the mean free energy per atom. Both monolayers exhibit only small fluctuations around their respective ground-state energies throughout the simulation. The insets show the relaxed supercell structures at the end of the simulation, confirming that no bond breaking or structural distortions occur, demonstrating the thermodynamical stability at room temperature.