Exotic $S=1$ spin-liquid state with fermionic excitations on the triangular lattice

Motivated by recent experiments on the material Ba${}_3$NiSb${}_2$O${}_9$, we consider a spin-one quantum antiferromagnet on a triangular lattice with the Heisenberg bilinear and biquadratic exchange interactions and a single-ion anisotropy. Using a fermionic “triplon” representation for spins, we study the phase diagram within mean-field theory. In addition to a fully gapped spin-liquid ground state, we find a state where one gapless triplon mode with a Fermi surface coexists with $d+id$ topological pairing of the other triplons. Despite the existence of a Fermi surface, this ground state has fully gapped bulk spin excitations. Such a state has linear in-temperature specific heat and constant in-plane spin susceptibility, with an unusually high Wilson ratio.

Figure 1

Schematic representation of the ground state in different limits of the Hamiltonian (1). The white arrows represent average spin; the arrows with disks indicate the director of the nematic order parameter. Details are discussed in the text.

Figure 2

The phase boundary between SL GSs with $p+ip$ and $d+id$ pairing. (a) The spin susceptibility ${\tilde{\chi }}_{xx}$ in the $d+id$ phase as a function of $D/J$ for $K/J=0.55$. The susceptibility is normalized by the average density of states $\overline{\nu }=({\nu }_x+{\nu }_z)/2$, where ${\nu }_x$ is calculated without the gap. (b) Gapped (dashed red line) and ungapped (solid blue line) Fermi surfaces of $x$, $y$, and $z$ fermions for $K/J=0.55$, $D/J=0.8$.