projective construction of spin nematic states in S=1/2 frustrated ferromagnets

An $SU(2)$ slave-boson formulation of bond-type spin nematic orders is developed in the context of quantum frustrated ferromagnets, where the spin nematic states are described as the resonating spin-triplet valence bond (RVB) states. Namely, the $d$-vector of the spin-triplet pairing ansatz plays the role of the so-called `director' in the spin nematic states. The low-energy excitations around such bond-type spin quadrupolar orders generally comprise the gauge boson, massless goldstone bosons, spinon individual excitations and their composites. Using the projective symmetry-group arguments, we will argue how to identify the number of massless gauge bosons. Applying this formulation, we will next enumerate possible `mixed' RVB ansatzes in the $S=\frac{1}{2}$ $J_1$-$J_2$ Heisenberg model on the square lattice ($J_1$ ferromagnetic nearest neighbor and $J_2$ antiferromagnetic next nearest neighbor), and argue their stability against gauge fluctuations. As a result, we found two stable ansatzes in the intermediate coupling region, $J_1:J_2=1:0.4$. One is the $Z_2$ `Balian-Werthamer (BW)' state stabilized by the Higgs mechanism. The other is the $SU(2)$ `chiral $p$-wave' state, where the massless gauge fluctuations are controlled by the Chern-Simon mechanism. Especially, the former $Z_2$ state exhibits the same spatial configuration of the spin quadrupolar moment as found in the previous exact diagonalization studies.