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 [O. N. Meetei, Phys. Rev. B 91, 054412 (2015)10.1103/PhysRevB.91.054412], 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≤λc1≈0.34J where J is the ferromagnetic exchange; and, (II) the S1 phase for λc1<λ≤λc2≈1.2J with emergent antiferromagnetic correlations. Both S1 and S2 phases are shown to exhibit power-law correlations, indicative of a gapless spectrum. Upon increasing λ>λc2 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 model in an external field that provides an approximate description of our model within mean-field theory.