  We study the ground state properties of the spin-1/2 frustrated ferromagnetic
J1-J2 Heisenberg model on the square lattice, employing projected BCS
wavefunctions with spin-triplet pairings of the spinon fields as trial
wavefunctions. Based on the variational Monte Carlo analysis, we argue that, in
the competing coupling regime, a certain type of the projected BCS
wavefunction, dubbed the projected Z2 planar state, achieves the best optimal
energy among the other competing states such as the ferromagnetic state and
collinear antiferromagnetic state. Like in quantum spin liquids, the projected
Z2 planar state preserves the translational symmetry of the square lattice.
However, it is also accompanied by a d-wave ordering of the quadrupole moments,
breaking the spin rotational symmetry. The state thus describes a quantum spin
analogue of the nematic liquid crystals. The calculated static correlation
functions also reveal that the projected Z2 planar state has a strong collinear
antiferromagnetic fluctuation.
