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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 $5d^4$ 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. We find two phases with increasing $λ$: (I) the S2 phase with two peaks in the structure factor for $λ\leλ_{c1}\approx 0.34 J$ where $J$ is the ferromagnetic exchange; and, (II) the $S1$ phase for $λ_{c1}<λ\leλ_{c2}\approx 1.2 J$ with emergent antiferromagnetic correlations. Both S1 and S2 phases are shown to exhibit power law correlations, indicative of a gapless spectrum. Upon increasing $λ> λ_{c2}$ leads to a product state of local spin-orbital singlets that exhibit exponential decay of correlations, indicative of a gapped phase. We obtain insights into the phases from the well-known Uimin-Lai-Sutherland (ULS) model in an external field that provides an approximate description of our model within mean field theory.