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 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 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 $\hat{x}$ direction and weakly coupled, three-dimensional QSH effect can be expected in these distorted materials, such as insulating compounds CeAs${}_{1+x}$Se${}_{1-y}$ and 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 ${\hat{s}}_x$ 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 $\ddot{I}{ƒ}_{yz}^x=(\frac{\hslash }{e})\frac{G_x{e}^2}{\ddot{I}€h}$ with $G_x$ a reciprocal lattice vector.