TABLE III: Comparison of spiral energies for different num- ber of variational parameters for 18 bosons on a 36-site clus- ter. The extended orbital of the spiral state has amplitudes 1, e i 2 π/ $^{3}$, and e i 4 π/ 3 for sites on sublattices A , B , and C respec- tively. The first spiral case has one nnb Jastrow pseudopo- tential which is taken to be the same between any pair of nnb sites, while the second spiral case has an additional parame- ter γ for the Huse-Elser phase factor. The third case has two more parameters w and p for a long range pseudopotential w | i − j | p between any pair of sites i and j . We also show trial energy for the Y state with 4 variational parameters, which has slightly higher energy than the spiral state. TABLE IV: Comparison of V perm energies for different num- ber of variational parameters for 11 bosons on a 36-site clus- ter. The V perm state is constructed by removing one boson from the uud state as described in the text, and the param- eters here are of the same type as in the uud case in Table I. We also show trial energy for the spiral state with 4 vari- ational parameters, which has higher energy than the V perm state. 4 serves as an additional variational parameter. For de- tails, the reader is referred to the original Ref. 22. Among our trial states, the Huse-Elser wavefunction has lower energy than the Y state for N$_{b}$ = 16 , . . ., 18, but only by a very small amount (cf. Fig. 4). Table III shows that the 18-boson spiral energy is only slightly lower than the Y energy. This is perhaps not surpris- ing since the classical 120 ◦ order may be viewed as the spiral or Y -shape order depending on the plane’s orien- tation. A recent variational study 24 using different con- structions of the spiral and Y states obtained − 0 . 1827 for their many-parameter spiral state, which is also lower than their Y trial energy by a small amount similar to that in our study. Other recent works 21,26 observed an abrupt change from the spiral to Y supersolid in the half- filled model as the spin anisotropy is varied through the SU(2)-invariant Heisenberg point. In principle, thinking in terms of wavefunctions, the spiral and Y can be dis- tinct phases with different postulated symmetry breaking also in the SU(2)-invariant model. However, this could also be a plane reorientation transition, and the closeness in energy of the Y trial states reflecting their ability to capture the 120 ◦ spiral order. Let us now consider densities slightly less than 1/3. We start from the uud state and picture it as a filled A sublattice. An appealing scenario is to introduce holes and let them move around on A and condense. We auto- matically retain charge order selecting the A sublattice vs B and C . The condensation of holes on the A gives boson superfluid order there and by proximity effect also on the B and C sublattices. Since t$_{AB}$ = t$_{AC}$ < 0, we expect the phase angle on the BC to be shifted by π from the A . The resulting supersolid is precisely the V state. Direct wavefunction implementation of this scenario is described in Appendix A and leads to a sum of perma- nents, which becomes prohibitively costly to evaluate for more than few holes. In the appendix, we also motivate a qualitatively similar wavefunction with a simpler am- plitude given by a single permanent, The first N$_{b}$ rows contain the localized orbitals of the uud construction evaluated at the boson positions, while the remaining N/ 3 − N$_{b}$ rows are filled with 1-s corre- sponding to “zero wavevector” condensate of holes (see Appendix A for details). Table IV shows the V$_{perm}$ energy for 11 bosons on the 6 × 6 cluster. Being a descendant of the excellent uud state, even with no Jastrow factor the V$_{perm}$ per- forms very well and lies roughly half-way between the ground state and the first excited state with the same quantum numbers (the latter is not listed in the ta- ble). In particular, the V$_{perm}$ clearly wins over the spi- ral superfluid with uniform density. Just as in the uud case, adding simple short-range Jastrow parameters fur- ther improves the trial energy of the V$_{perm}$ state. At this stage, we did not include long-range Jastrow pseu- dopotentials, which would be needed for correct long- wavelength description 29 of superfluid correlations in the V phase. The V$_{perm}$ state gives our best variational energies for N$_{b}$ = 6 , . . . , 11. In Appendix B, we measure properties of this state and verify the superfluid order with opposite signs on the A and BC sublattices as anticipated above. D. V state at n < ∼ 1 / 3 〈{ r$_{k}$ } | ψ$_{V}$$_{perm}$ 〉 = Perm     φ loc 1 ( r$_{1}$ ) . . . φ loc N/ $_{3}$( r$_{1}$ ) . . . . . φ loc 1 ( r$_{N}$$_{b}$ ) . . . φ loc N/ $_{3}$( r$_{N}$$_{b}$ ) 1 . . . 1 . . . . . 1 . . . 1    $^{.}$$^{(12)}$ Trial state N orb par N Jas short N Jas long N tot par E/bond classical -0.1250 Y 1 1 2 4 -0.1774 spiral 0 1 0 1 -0.1728 0 1 0 2 -0.1791 0 1 2 4 -0.1795 exact -0.1868 Trial state N orb par N Jas short N Jas long N tot par E/bond classical -0.0683 spiral 0 1 2 4 -0.1081 V perm 1 0 0 1 -0.1141 1 1 0 2 -0.1153 1 2 0 3 -0.1154 exact -0.1161