Spin Liquid Phases for Spin-1 Systems on the Triangular Lattice

Motivated by recent experiments on material ${\mathrm{Ba}}_3{\mathrm{NiSb}}_2{\mathrm{O}}_9$, we propose two novel spin liquid phases ($A$ and $B$) for spin-1 systems on a triangular lattice. At the mean field level, both spin liquid phases have gapless fermionic spinon excitations with quadratic band touching; thus, in both phases the spin susceptibility and $\gamma =C_v/T$ saturate to a constant at zero temperature, which are consistent with the experimental results on ${\mathrm{Ba}}_3{\mathrm{NiSb}}_2{\mathrm{O}}_9$. On the lattice scale, these spin liquid phases have $\mathrm{Sp}(4)\sim \mathrm{SO}(5)$ gauge fluctuation, while in the long wavelength limit this Sp(4) gauge symmetry is broken down to $\mathrm{U}(1)\times Z_2$ in the type $A$ spin liquid phase, and broken down to $Z_4$ in the type $B$ phase. We also demonstrate that the $A$ phase is the parent state of the ferroquadrupole state, nematic state, and the noncollinear spin density wave state.

Figure 1

(a) The spin liquid we are considering contains a quadratic band touching at $\overrightarrow{k}=0$ (hexagon), and Dirac points (squares) at the corners of the Brillouin zone. (b) With a nonzero and small nematic order $N_1>0$, the quadratic band touching is split into two Dirac points, and the locations of the other Dirac points are shifted.