Long-range spin-orbital order in the spin-orbital SU(2)×SU(2)×U(1) model

By using the tensor-network state algorithm, we study a spin-orbital model with $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$ symmetry on the triangular lattice. This model was proposed to describe some triangular ${d}¹$ materials and was argued to host a spin-orbital liquid ground state. In our work the trial wave function of its ground state is approximated by an infinite projected entangled simplex state and optimized by the imaginary-time evolution. Contrary to the previous conjecture, we find that the two SU(2) symmetries are broken, resulting in a stripe spin-orbital order with the same magnitude $m=0.085(10)$. This value is about half of that in the spin-1/2 triangular Heisenberg antiferromagnet. Our result demonstrates that although the long-sought spin-orbital liquid is absent in this model the spin-orbital order is significantly reduced due to the enhanced quantum fluctuation. This suggests that high-symmetry spin-orbital models are promising in searching for exotic states of matter in condensed-matter physics.