Magnetic phase transitions in quantum spin-orbital liquids

We investigate the spin and orbital correlations of a superexchange model with spin S=1 and orbital L=1 relevant for 5d⁴ transition metal Mott insulators, using exact diagonalization and density matrix renormalization group (DMRG). For spin-orbit coupling λ=0, the orbitals are in an entangled state that is decoupled from the spins, leading to emergent spin-orbital separation within spin-orbital interacting system. We find two phases with increasing λ: (I) the S2 phase with two peaks in the structure factor for λ < λ_c1 ≈ 0.34 J, where J is the ferromagnetic exchange, and (II) the S1 phase λ_c1 < λ < λ_c2 ≈ 1.2 J with antiferromagnetic correlations. The λ = 0, S2 and S1 phases are shown to exhibit power law correlations, indicative of a gapless phase. Increasing λ > λ_c2 leads to a product state of local spin-orbital singlets that exhibits exponential decay of correlations, indicative of a gapped phase. Using mean-field like approximation, we demonstrate that our model can be approximated with the well-known Uimin-Lai-Sutherland (ULS) model with an external field.