5 TABLE V: Comparison of V$_{Bijl}$$_{−}$$_{Jastrow}$ trial energies for dif- ferent number of variational parameters for 3 bosons on a 36-site cluster. The extended orbital of the V$_{Bijl}$$_{−}$$_{Jastrow}$ state has amplitude e µ/ 2 on sublattice A and − 1 on sublattices B and C ; µ is the only variational parameter in the first listed V$_{Bijl}$$_{−}$$_{Jastrow}$ case. The second V$_{Bijl}$$_{−}$$_{Jastrow}$ case has one nnb Jastrow pseudopotential which is taken to be the same be- tween any pair of nnb sites, while the third case has two ad- ditional parameters for a long range pseudopotential which is the same as in the spiral case in Table III. We also show trial energy for the spiral state with 4 variational parameters, which has slightly higher energy than the V$_{Bijl}$$_{−}$$_{Jastrow}$ state. Trial state N orb par N Jas short N Jas long N tot par E/bond classical 0.13542 spiral 0 1 2 4 0.12885 V BJ 1 0 0 1 0.13216 1 1 0 2 0.12913 1 1 2 4 0.12869 exact 0.12845 E. V state at n ≪ 1 / 3 This orbital has opposite signs on the A and BC sublat- tices as expected from Fig. 3d. The “chemical potential” µ on the A sublattice allows us to control the charge or- der. Similar to other wavefunctions for states with super- fluid order, it is necessary to include a long-range Jastrow pseudopotential. As we argue below, this wavefunction is a natural can- didate at low boson densities. On the 6 × 6 cluster, it optimizes better than the V$_{perm}$ state for N$_{b}$ ≤ 6 and also has better energy than the spiral state, see Fig. 4. As an example of variational results, Table V shows the V$_{Bijl}$$_{−}$$_{Jastrow}$ energy for 3 bosons on the 6 × 6 cluster. We see that the V state is slightly better than the spiral state. However, both states are quite close in energy and close to the exact ground state. We discuss this more below and see what we can infer about the competition between the coplanar and spiral states from ED spectroscopy. First, we want to connect the competing V$_{Bijl}$$_{−}$$_{Jastrow}$ and spiral states with physics at low boson densities. In the absence of interaction, the kinetic energy mini- mizes at two distinct points in the Brillouin zone, ± ⃗ Q = ± (4 π/ 3 , 0). Boson condensation at one point gives rise to the spiral phase; schematically, the spiral wavefunc- The above wavefunction for the V phase is obtained from the strong uud state and a priori is not expected to remain good at low density. Here we consider an alterna- tive construction of the V supersolid using Bijl-Jastrow- type wavefunction, working directly with bosons and con- densing them into an appropriate extended orbital, | ψ$_{V,}$$_{Bijl}$$_{−}$$_{Jastrow}$ 〉 = ( ∑ r φ ext V ( r ) b † r ) N$_{b}$ | 0 〉 (13) φ ext V ( r ) = { e µ/ $^{2}$, r ∈ A − 1 , r B, C (14) { ( b † ⃗ $_{Q}$) $^{3}$, ( b † − ⃗ $_{Q}$) $^{3}$, ( b † ⃗ $_{Q}$) $^{2}$b † − ⃗ $_{Q}$, b † ⃗ $_{Q}$( b † − ⃗ $_{Q}$) 2 } . (15) Our spiral wavefunction construction gives essentially the first two states with momentum quantum number ⃗ k = 0. Our 3 degenerate V -states, upon making combinations that are momentum eigenstates, give an even-parity ⃗ k = 0 combination as well as the last two states with ⃗ k = ± ⃗ Q from Eq. (15). Finally, the 3 degenerate Ψ-type states give an odd-parity ⃗ k = 0 combination and the same two ⃗ k = ± ⃗ Q states. It is clear that these trial states are not independent for this small number of bosons; we cannot resolve the phases, but we can start looking for some tendencies. For example, we can view the fact that the ⃗ k = ± ⃗ Q are lower in energy than ⃗ k = 0 as an indication for the coplanar states being better than the spiral. In principle, we could also try to resolve between the V and Ψ by comparing the even/odd-parity ⃗ k = 0 states, but the splitting is too tiny. We have similarly examined ED spectra with N$_{b}$ = 4 , . . . , 9 bosons, paying attention to near degeneracy of ground states and their quantum numbers. The resolu- tion between the spiral and coplanar states due to inter- actions becomes clearer with increasing density, and in each instance the ED data is consistent with the copla- nar states being better, which is also supported by the VMC data. As far as the resolution between the V and Ψ states is concerned, we can not tell anything with bo- son number below 6, while for higher boson density we start seeing evidence in favor of the V state. The V state is expected coming from the n = 1 / 3 plateau as Returning to our example with 3 bosons on the 6 × 6 lattice, the ED ground state has momentum quantum number ⃗ k = ⃗ Q (there is also a degenerate state with opposite momentum), while we also find two very close states with ⃗ k = 0, one with even and the other with odd parity under inversion. This can be traced to 4 degener- ate eigenstates of the kinetic energy, tion is given by ( b † ⃗ $_{Q}$) $^{N$_{b}$}$| 0 〉 [or ( b † − ⃗ $_{Q}$) $^{N$_{b}$}$| 0 〉 for the oppo- site wavevector]. A more complex condensation pattern including both points produces a coplanar state, with schematic wavefunction ( e $^{iα}$b † ⃗ $_{Q}$+h . c . ) $^{N$_{b}$}$| 0 〉 . When α = 0, this gives boson orbital φ ( r ) = cos( ⃗ Q · ⃗ r ) taking values { +1 , − 1 / 2 , − 1 / 2 } on the three sublattices, which is es- sentially the φ ext V orbital in Eq. (14). On the other hand, α = π/ 2 corresponds to a different state with zero bo- son density on one sublattice and alternating superfluid phases on the other two sublattices; in terms of spins, this is a coplanar “Ψ”-type state which has similar sym- metry to the Y state in Fig.2b, but with the vertical spin flipped up. For either V or Ψ, there are two more degen- erate states given by lattice translations or equivalently by adding ± 2 π/ 3 to the phase α . Reference 11 studies the dilute boson problem analytically for the isotropic lattice and predicts that four-boson interactions select coplanar states; this study does not resolve between the V and Ψ states, which requires considering six-boson terms.