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

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 CaMnSb2. In this paper, we demonstrate that the Jahn-Teller effect enlarges the nontrivial SOC gap in the distorted structure, e.g., LaAsS and SrZnSb2. 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 (x) over cap direction and weakly coupled, three-dimensional QSH effect can be expected in these distorted materials, such as insulating compounds CeAs1+xSe1-y and EuCdSb2. 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 Cs+ = 2 when there is a gap between two sets of (s) over cap (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) = (h/e) G(x)e(2)/pi h with G x a nh reciprocal lattice vector.