11 other densities in the highly anisotropic limit. For exam- ple, for 7 bosons on the 36-site cluster, one of the chains now contains two bosons, and the ground state of the decoupled chains Hamiltonian is 6-fold degenerate due to 6 possible ways of choosing this chain. The finite-size gaps “protect” this situation until the interchain coupling J ′ becomes sufficiently large. Such observations on the ED spectra show serious limitations of the small system study in the anisotropic model. Going over all ED data for N$_{b}$ ≤ 12, we conclude that δ > ∼ 0 . 5 regime can be ra- tionalized as such weakly-coupled finite chains, with no clear resolution of the ultimate state. This is labeled as “quasi-1d” region in Fig. 9. One of the goals of the 6 × 6 study was to have ED reference for our trial states. Having achieved some con- fidence in the good energetics of these states (despite their limitations), we now want to discuss variational re- sults for larger sizes. Specifically, on the 36-site cluster, we have not considered the possibility of incommensu- rate V state: while we know how to accommodate the incommensurate spiral state, we do not have similar con- struction for the incommensurate coplanar state. On the 36-site cluster, we see that incommensuration becomes important for δ ≥ 0 . 2. In fact, as we discuss below, we think that for this density the V state is probably in- commensurate already for smaller anisotropy, but also extends to larger anisotropy in the competition against the spiral. In this section, we focus on the high-field regime where the incommensurate V and spiral are the main competing candidates. First, we briefly describe the relevant physi- cal picture. Beginning with the near-saturation limit, we consider a gas of free bosons hopping on the triangular lattice with the following kinetic energy spectrum: The band minima occur at ⃗ Q = ± ( Q$_{x}$, 0) with A condensation of bosons at these points gives rise to a degenerate manifold of states spanned by At low densities, the degeneracy is lifted by nearest neigh- bour repulsion. To see how this happens, we expand the interaction in terms of the two dominant spectral modes Note that for incommensurate ⃗ Q the relative phase be- tween b † ⃗ Q and b † − ⃗ Q is not fixed, which we indicated with α . This is not important in an infinite system since ⃗ Q · ⃗ r To address the competition between these two phases quantitatively at finite density, we implement a varia- tional study on m × n rectangular clusters such that a fitting wavevector Q$_{x}$ = 2 πp/m is close to the spiral wavevector at each anisotropy (with appropriate integers p ). For the spiral phase, we use the same anisotropic wavefunction described earlier. A candidate wavefunc- tion for the incommensurate V phase is constructed as follows: From the positive sign in the second term, which dislikes the V , we may expect the boundary between the V and spiral phases to shift in favor of the spiral phase as the density increases. and then replace the operators by c-numbers $^{11,12}$: The effect of nearest neighbour repulsion is determined by the sign of υ . For J $^{′}$/J < 0 . 39, υ is positive and the incommensurate spiral (e.g., | b$_{⃗}$ Q | = 0 and | b$_{−}$$_{⃗}$ Q | = 0) wins. For 0 . 39 < J $^{′}$/J ≤ 1 . 59, υ is negative and the incommensurate V ordering ( | b$_{⃗}$ Q | = | b$_{−}$$_{⃗}$ Q | ) becomes more stable. The prediction from this approximate treatment is consistent with the spiral phase found in the nearly decoupled chains limit near saturation and in the highly anisotropic dilute boson study for the Cs$_{2}$CuCl$_{4}$; 10,12 this is also consistent with the coplanar phase found in the isotropic dilute boson study. 11 In the above discussion, we have neglected the effect of hard core interaction. Intuitively, this should be more im- portant at higher density: The role of the hard core con- straint is to prevent two bosons already in nearest neigh- bour contact from further occupying the same site, while at low density such contacts are avoided due to nearest neighbor repulsion. To see whether the hard core inter- action favors the V or spiral phase, we expand the on-site repulsion energy in terms of the two spectral modes: E. Incommensurate V versus spiral study at low to intermediate boson densities ϵ$_{k}$ = J cos( k$_{x}$ ) + 2 J $^{′}$cos ( k$_{x}$ 2 ) cos ( √$_{3}$ k$_{y}$ 2 ) . (25) Q$_{x}$ = 2 acos( − J $^{′}$/ 2 J ) . (26) { ( b † ⃗ $_{Q}$)$^{m}$( b † − ⃗ $_{Q}$) N$_{b}$ − $^{m}$| 0 〉 ; m = 0 , 1 , . . . , N$_{b}$ } . (27) | ψ$_{V}$$_{in}$ 〉 = ( e $^{iα}$b † ⃗ Q + e − $^{iα}$b † − ⃗ Q ) N$_{b}$ | 0 〉 = ∑ r cos( ⃗ Q · ⃗ r + α ) b † r ) N$_{b}$ | 0 〉 . (33) ( b † $_{r}$b$_{r}$ ) 2 ∼ ( | b$_{⃗}$ Q | 2 + | b$_{−}$$_{⃗}$ Q | 2 ) 2 + 2 | b$_{⃗}$ Q | 2 | b$_{−}$$_{⃗}$ Q | $^{2}$. (32) b$_{r}$ ∼ e i ⃗ Q · ⃗ r b$_{⃗}$ Q + e − i ⃗ Q · ⃗ r b$_{−}$$_{⃗}$ $_{Q}$, (28) ˆ H$_{int}$ = ∑ 〈 rr ′ 〉 J$_{rr}$ ′ b † $_{r}$b$_{r}$ b † r ′ b$_{r}$ ′ (29) ∼ ( J + 2 J $^{′}$) ( | b$_{⃗}$ Q | 2 + | b$_{−}$$_{⃗}$ Q | 2 ) 2 + 2 υ | b$_{⃗}$ Q | 2 | b$_{−}$$_{⃗}$ Q | $^{2}$, (30) υ = J cos(2 Q$_{x}$ ) + 2 J $^{′}$cos( Q$_{x}$ ) . (31)