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 $5{d}^{4}$ transition-metal Mott insulators [O. N. Meetei et al., Phys. Rev. B 91, 054412 (2015)], using exact diagonalization and density matrix renormalization group (DMRG). For spin-orbit coupling $\ensuremath{\lambda}=0$, the orbitals are in an entangled state that is decoupled from the spins. We find two phases with increasing $\ensuremath{\lambda}$: (I) the $S2$ phase with two peaks in the structure factor for $\ensuremath{\lambda}\ensuremath{\le}{\ensuremath{\lambda}}_{c1}\ensuremath{\approx}0.34J$ where $J$ is the ferromagnetic exchange; and, (II) the $S1$ phase for ${\ensuremath{\lambda}}_{c1}l\ensuremath{\lambda}\ensuremath{\le}{\ensuremath{\lambda}}_{c2}\ensuremath{\approx}1.2J$ with emergent antiferromagnetic correlations. Both $S1$ and $S2$ phases are shown to exhibit power-law correlations, indicative of a gapless spectrum. Upon increasing $\ensuremath{\lambda}g{\ensuremath{\lambda}}_{c2}$ 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 model in an external field that provides an approximate description of our model within mean-field theory.