Based on the variational Monte Carlo analysis, I first argue that a projected BCS wavefunction with spin-triplet pairing of the spinon field, dubbed as the projected Z2 planar state, achieves the best optimal energy among the other competing phases such as the ferromagnetic state and collinear Neel state. Like in quantum spin liquids, this wavefunction preserves the translational symmetry of the square lattice, indicating its spin-liquid like character. However, the state is also shown to be accompanied by the `d-wave' ordering of the magnetic quadrupole moment, breaking the global spin-rotational symmetry. Having both spin-liquid character and spontaneous symmetry breaking (SSB) phase character, this projected Z2 planar state can be regarded as a quantum spin analogue of `liquid-crystal' like state. I examine the symmetry properties and the static spin correlation function of this projected BCS wavefucntion, only to compare them with those suggested by the previous exact diagonalization studies. The comparison indicates that the state is indeed realized in a certain range of the intermediate coupling regime of the present spin model.

Motivated by this consistency, I next discuss the dynamical spin structure factor in the present quantum spin nematic phase. Specifically, using a standard large-N loop expansion, I take into account the fluctuation around the mean-field Z2 planar state within the one-loop level.

The spin structure factor thus obtained has two aspects; spin-liquid like character and SSB phase like character. The former feature manifests itself as (i) the coherent peak of a gapped `gauge' boson and (ii) the Stoner continuum associated with the individual excitations of gapped free spinons. On the one hand, the SSB phase character is represented by the gapless spin-wave modes, whose spectral weight vanish as a linear function of the momentum near its gapless point. The latter feature especially suggests that, as in conventional site-type spin nematc phase, the inverse of the longitudinal relaxation time of NMR experiments in this ordered phase vanishes as a cubic function in temperature, 1/T_1 ~ T^3.

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