Twisted nodal wires and three-dimensional quantum spin Hall effect in distorted square-net compounds

Abstract

Recently, square-net materials have attracted lots of attention for the Dirac semimetal phase with negligible spin-orbit coupling (SOC) gap, e.g., ZrSiS/LaSbTe and ${\mathrm{CaMnSb}}_{2}$. In this paper, we demonstrate that the Jahn-Teller effect enlarges the nontrivial SOC gap in the distorted structure, e.g., LaAsS and ${\mathrm{SrZnSb}}_{2}$. Its distorted $X$ square-net layer ($X=$ P, As, Sb, Bi) resembles a quantum spin Hall (QSH) insulator. Since these QSH layers are simply stacked in the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x}$ direction and weakly coupled, three-dimensional QSH effect can be expected in these distorted materials, such as insulating compounds ${\mathrm{CeAs}}_{1+x}{\mathrm{Se}}_{1\ensuremath{-}y}$ and ${\mathrm{EuCdSb}}_{2}$. Our detailed calculations show that it hosts two twisted nodal wires without SOC [each consists of two noncontractible time-reversal symmetry- and inversion symmetry-protected nodal lines touching at a fourfold degenerate point], while with SOC it becomes a topological crystalline insulator with symmetry indicators $(000;2)$ and mirror Chern numbers $(0,0)$. The nontrivial band topology is characterized by a generalized spin Chern number ${C}_{s+}=2$ when there is a gap between two sets of ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{s}}_{x}$ eigenvalues. The nontrivial topology of these materials can be well reproduced by our tight-binding model and the calculated spin Hall conductivity is quantized to ${\ensuremath{\sigma}}_{yz}^{x}=(\frac{\ensuremath{\hbar}}{e})\frac{{G}_{x}{e}^{2}}{\ensuremath{\pi}h}$ with ${G}_{x}$ a reciprocal lattice vector.