The origin of anomalous 3rd neighbor exchange in 2D triangular magnets (NiGa2S4 and others)

2D magnetic materials with triangular lattices have been attracting much interest. Among them one finds the parent compound of an exotic superconductor, Na$_x$CoO$_2\cdot y$H$_2$O, A-type antiferromagnets like NaNiO$_2$, in-plane antiferromagnetism (LiCrO$2$), spin-liquid type materials (NiGa$_2$S$_4$), charge-order (AgNiO$_2$). The main structural motif in all of them is the AB$_2$ plane, where A is a transition metal and B is oxygen or sulfur. Experiments and calculations inevitably find anomalously strong 3rd neighbor exchange coupling in all these triangular planes, despite different band fillings and different magnetic ground states. I will explain why this happens, why this effect is so universal, and why it can be understood entirely on a one-electron level. I will use as an example NiGa$_2$S$_4$, with a reference to Na$_x$CoO$_2$ as well.