10 that only extend to nearest neighbour sites, it fails to capture longer range correlations in the chain direction expected in a more anisotropic system. We would need to use more extended orbitals in the permanent, but we have not pursued this. On the other hand, the 2-parton realization readily accommodates the lattice anisotropy via the parton hoppings, Eq. (23), and provides a sim- ple way to continue our study of the uud state to larger anisotropy. In the highly anisotropic region, we obtain good trial energies for the 6 × 6 system using the 2-parton wavefunc- tion without the chemical potential. However, if we con- sider the low energy cutoff due to the finite cluster size, it is clear that the study cannot resolve the true phase in the thermodynamic limit. Specifically, in the decoupled chains limit, we obtain a very accurate wavefunction for two bosons on a 6-site chain by using antiperiodic bound- ary conditions for the partons, cf. Eq. (24). The corre- sponding parton spectrum nicely accommodates two par- ticles and has a large finite-size gap to next levels, which persists up to moderate interchain couplings. While our 2-parton state by virtue of good fluxes naturally builds in good interchain exchange correlations, we cannot resolve the thermodynamic phase (e.g., the development of the √$_{3}$ × √$_{3 CDW) if the relevant energy scale is much lower}$ than the finite-size gap. To determine how far the uud phase might extend into the anisotropic region, we repeat the 2-parton calculation on a large 24 × 24 cluster and find that the √$_{3}$ × √$_{3}$ chemical potential surprisingly remains non-zero up to δ ≈ 0 . 7. We also check that the parton spectrum for the optimal parameters is fully gapped and is connected to the strongly gapped CDW limit, so the trial wavefunction is indeed a valid charge-ordered Mott insulator of bosons as discussed in Sec. II H. We thus conclude that the uud state persists to rather strong anisotropy, albeit the CDW order becomes progressively weaker. Interestingly, Ref. 10 would predict the same √$_{3}$ × √$_{3 CDW order in}$ the nearly decoupled chains limit at density n = 1 / 3. Combining with our variational work, this suggests that the uud phase may in fact extend continuously from δ = 0 up to δ = 1 (See Fig. 10). A rigorous confrontation to this conjecture could be provided, for example, by a systematic DMRG study of 3 × L ladders at density n = 1 / 3 varying J $^{′}$/J from 1 to 0 and monitoring the evolution of the √$_{3}$ × √$_{3 charge order.}$ While our study agrees with the observed plateau in Cs$_{2}$CuBr$_{4}$ ( δ ≈ 0 . 3), it contradicts the absence of the plateau in Cs$_{2}$CuCl$_{4}$ ( δ ≈ 0 . 66). It is likely that residual interactions (e.g. such as Dzyaloshinskii-Moriya) have to be added to the Heisenberg model in order to describe the latter material, 10 and they can change the energetics balance against the (very weak) uud state in this highly anisotropic system. We can similarly rationalize all our ED observations at In the highly anisotropic region, the figure shows re- markable agreement between the ED and the 2-parton energies, where we impose uniform π/ 6 and 5 π/ 6 flux per triangle for the d$_{1}$ and d$_{2}$ partons respectively (Chern- Simons state described in Sec. II H). Despite the ex- cellent agreement, we simply conclude that the highly anisotropic region is strongly dominated by quasi-1d physics and finite size effects. Specifically, in the ED calculation on the 36-site cluster with 6 bosons, we find a non-degenerate ground state and a relatively large ex- citation energy gap. We interpret this as follows. In the decoupled chains limit, each chain contains one boson; for such a segment of length L = 6, one expects a non- degenerate ground state with a large excitation gap due to finite size. This gap persists as the chains are coupled, particularly because of some frustration present in the triangular lattice geometry. Figure 7 shows the trial energies of commensurate V$_{Bijl}$$_{−}$$_{Jastrow}$ , incommensurate spiral [treated as described around Eq. (21)], and 2-parton wavefunctions for 6 bosons on the 6 × 6 cluster. At low anisotropy, the V$_{Bijl}$$_{−}$$_{Jastrow}$ state is a good candidate. For increas- ing anisotropy, a change to the incommensurate spiral is observed, which eventually loses to the “quasi-1d” phase represented by the 2-parton wavefunction with zero chemical potential. D. n = 1 / 6 FIG. 7: Comparison of spiral, commensurate V$_{Bijl}$$_{−}$$_{Jastrow}$ , and 2-parton trial energies as a function of anisotropy for 6 bosons on the 6 × 6 cluster. The E 2 p values are obtained with higher chemical potential on one sublattice for δ ≤ 0 . 3 (in agreement with this state trying to capture 3-sublattice features near small anisotropy in this small sample). We also performed a much larger study at n = 1 / 6 comparing commensurate and incommensurate V states, incommensu- rate spiral, and 2-parton states; from this study, the region of commensurate V is actually quite small, while the incom- mensurate V dominates over the spiral over a range δ ≤ 0 . 3 (see text for more details). 0 1 2 3  0  0.2  0.4  0.6  0.8  1 (E-E ED )/10 -2 J  = (J-J’)/J Spiral V 2p ( >0) 2p ( =0)