BCS-BEC crossover in a (t2g)4 excitonic magnet

The condensation of spin-orbit-induced excitons in ${t}_2g⁴$ electronic systems is attracting considerable attention. At large Hubbard $U$, antiferromagnetism was proposed to emerge from the Bose-Einstein Condensation (BEC) of triplons (${J}_{\text{eff}}=1$). Here, we show that even at 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 ($\ensuremath{\lambda}$) in one dimension using the density matrix renormalization group, while in two dimensions we use the Hartree-Fock approximation (HFA). Employing these techniques, we provide the full $\ensuremath{\lambda}$ versus $U$ phase diagrams for both one- and two-dimensional lattices. Our main result is that at intermediate Hubbard $U$, increasing $\ensuremath{\lambda}$ at fixed $U$ the system transitions from an incommensurate spin-density-wave metal to a Bardeen-Cooper-Schrieffer (BCS) excitonic insulator, with coherence length ${r}_{\mathrm{coh}}$ of $\mathcal{O}(a)$ and $\mathcal{O}(10a)$ in $1d$ and $2d$, respectively, with $a$ being the lattice spacing. Further increasing $\ensuremath{\lambda}$, the system eventually crosses over to the BEC limit (with ${r}_{\mathrm{coh}}\ensuremath{\ll}a$).