14 ther V or spiral has lower energy. Our earlier uud study showed that the 2-parton construction can also be useful for studying CDW phases; however, similar implemen- tations at the above densities again failed to reveal any stable charge ordering. This suggests that any such or- der, if present at all, is likely to be very weak. One of the findings from our study is that for the Cs$_{2}$CuBr$_{4}$ anisotropy, the system is in the incommensu- rate coplanar phase close to the saturation fields, 37 and there may be a transition to the non-coplanar spiral state at lower fields; this could be responsible for one of the fea- tures in the Cs$_{2}$CuBr$_{4}$ experiment. We cannot exclude other more complex cascade of phases. Furthermore, ad- ditional residual interactions not treated here may be im- portant for understanding the phases of Cs$_{2}$CuBr$_{4}$ in the field. This remains a fascinating open problem. 〈 η | Ψ 〉 = ∑ ′ R$_{1}$...R$_{N}$$_{h}$ φ$_{h}$ ( R$_{1}$ ) . . . φ$_{h}$ ( R$_{N}$$_{h}$ ) 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}$ ) φ loc 1 ( R$_{1}$ ) . . . φ loc N/ $_{3}$( R$_{1}$ ) . . . . . φ loc 1 ( R$_{N}$$_{h}$ ) . . . φ loc N/ $_{3}$( R$_{N}$$_{h}$ )     ≈ 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}$ ) c$_{1}$ . . . c$_{N/}$$_{3}$ . . . . . c$_{1}$ . . . c$_{N/}$$_{3}$     (A3) The primed sum indicates that R$_{1}$, . . . , R$_{N}$$_{h}$ need to be different from each other and from all r$_{1}$, . . . , r$_{N}$$_{b}$ . Close to the 1 / 3 plateau, the density of holes is small, and we can approximately replace the restricted sum by an unrestricted sum. Performing independent summations over R$_{1}$, . . . , R$_{N}$$_{h}$ gives the last expression in the form of a single permanent, where is an “overlap” of the φ$_{h}$ and φ loc j orbitals. On physics grounds, the hole orbital φ$_{h}$ needs to respect the symme- tries of the uud state ( φ$_{h}$ = const over the A sublattice). In this case, c$_{j}$ is independent of j , and up to a nor- malization constant we can replace all matrix elements in the last N$_{h}$ rows by 1. The approximate single per- manent form is a valid variational wavefunction by itself, and this is the state we use in the main text and call V$_{perm}$ . We calculate the order parameters and correlation functions in the permanent-type trial states, since we do not have much experience with such wavefunctions Appendix B: Correlation functions of permanent-type states c$_{j}$ = ∑ R φ$_{h}$ ( R ) φ loc j ( R ) (A4) Appendix A: Motivation for V perm wavefunction for n < ∼ 1 / 3 , Eq. (12) We begin with the uud state with N/ 3 bosons, Eq. (6), and put N$_{h}$ = N/ 3 − N$_{b}$ holes in a “hole orbital” φ$_{h}$ ( R ), For a boson configuration we obtain an amplitude, | Ψ 〉 = ( ∑ R φ$_{h}$ ( R ) b$_{R}$ ) N$_{h}$ | ψ$_{uud}$ 〉 . (A1) | η 〉 = b † r$_{1}$ . . . b † r$_{N}$$_{b}$ | 0 〉 , (A2) for Mott insulators or supersolids. The non-permanent V and spiral trial states are more obvious constructions, and therefore omitted. For Y , uud , and V$_{perm}$ trial states, a three sublat- tice modulation is observed in the CDW order param- eter 〈 n$_{r}$ 〉 . As expected, the density structure factor 〈 n$_{−}$$_{q}$n$_{q}$ 〉 reveals sharp peaks near the reciprocal vectors ⃗ Q = ± ( 4 π 3 , 0) for all three trial states. Figure 11 shows the correlation functions 〈 b † $_{r}$b$_{r}$ ′ 〉 between two sites for these trial states. The uud state has rapidly decaying correlations between any two sites, which is expected in this Mott insulator state. For the Y supersolid state, the correlations decay rapidly when at least one site lies on the higher density sublattice A (i.e., as if this sublattice is Mott-insulating), while they are long-ranged when both sites reside on the BC honeycomb sublattice. The signs of the correlations are positive for all pairs of B - B or C - C sites, and negative for all B - C pairs, which is consistent with the Y spin order shown in Fig. 2. Finally, for the V$_{perm}$ trial state, long-ranged correlation exists between any two sites on the lattice. The signs are negative for all pairs of A - B or A - C sites, and positive for all B - C pairs (as well as A - A , B - B , and C - C pairs), which is consistent with the V spin order shown in Fig. 2. Thus, we have verified our intuition about the physical properties of these states.