  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 $\sigma^{x}_{yz} =
(\frac{\hbar}{e})\frac{G_xe^2}{\pi h}$ with $G_x$ a reciprocal lattice vector.

Materials Science