BCS-BEC crossover in a $(t_{2g})^4$ Excitonic Magnet

The condensation of spin-orbit-induced excitons in $(t_{2g})^4$ electronic systems is attracting considerable attention. In the large Hubbard U limit, antiferromagnetism was proposed to emerge from the Bose-Einstein Condensation (BEC) of triplons ($J_{\textrm{eff}} = 1$). In this publication, we show that even for the weak and intermediate U regimes, the spin-orbit exciton condensation is possible leading also to staggered magnetic order. The canonical electron-hole excitations (excitons) transform into local triplon excitations at large U , and this BEC strong coupling regime is smoothly connected to the intermediate U excitonic insulator region. We solved the degenerate three-orbital Hubbard model with spin-orbit coupling ($\lambda$) in one-dimensional geometry using the Density Matrix Renormalization Group, while in two-dimensional square clusters we use the Hartree-Fock approximation (HFA). Employing these techniques, we provide the full $\lambda$ vs U phase diagrams for both one- and two- dimensional lattices. Our main result is that at the intermediate Hubbard U region of our focus, increasing $\lambda$ at fixed U the system transitions from an incommensurate spin-density-wave metal to a Bardeen-Cooper-Schrieffer (BCS) excitonic insulator, with coherence length r coh of O(a) and O(10a) in 1d and 2d, respectively, with a the lattice spacing. Further increasing $\lambda$, the system eventually crosses over to the BEC limit (with r coh << a).