Insulating Ferromagnetism in La$_{4}$Ba$_{2}$Cu$_{2}$O$_{10}$: an \textit{Ab Initio} Wannier Function Analysis

Microscopic mechanisms of the puzzling insulating ferromagnetism of half-filled La$_{4}$Ba$_{2}$Cu$_{2}$O$_{10}$ are elucidated with energy-resolved Wannier states. The dominant magnetic coupling, revealed through evaluated parameters ($t$, $U$, and $J$), turns out to be the intersite direct exchange, a currently ignored mechanism that overwhelms the antiferromagnetic superexchange. By contrast, the isostructural Nd$_{4}$Ba$_{2}$Cu$_{2}$O$_{10}$ develops the observed antiferromagnetic order via its characteristics of a 1D chain. Surprisingly, the in-plane order of both cases is \textit{not} controlled by coupling between nearest neighbors. An intriguing pressure-induced ferromagnetic to antiferromagnetic transition is predicted.