Chern non-trivial two-dimensional square-net nodal line semimetal
We investigate a two-dimensional $p_xp_y$ square-net tight-binding model with two sublattices and introduce a symmetry-informed, momentum-dependent spin-orbit coupling (SOC) with a finite-range real-space hopping representation. Its two components jointly gap the nodal line, while one breaks time-reversal symmetry, producing a Chern-insulating phase with an occupied band Chern number $C=2$. Nanoribbon calculations reveal edge-localized states traversing this gap, with two net chiral modes per edge. We further show that the conserved spin sectors are unitarily equivalent and therefore have equal Chern numbers, ruling out a nontrivial time-reversal-protected $\mathbb{Z}_2$ phase within this ansatz. Our results establish a phenomenological route to Chern topology in square-net two-dimensional systems and clarify the symmetry constraints governing the accessible insulating phases.