12 FIG. 8: Variational phase boundary between V and spiral (S) phases obtained from calculations on cluster sizes 18 × 18, 51 × 20, 27 × 20, 44 × 20, 47 × 20, 38 × 10, 39 × 10, 53 × 10, 60 10, and 99 × 10, where the corresponding anisotropy increases from 0 to 0.9 in equal intervals; the sizes are chosen so as to best accommodate the classical wavevector, Eq. (26). The V phase remains commensurate at δ = 0 . 1 for n ≥ 1 / 5. Note that at high anisotropy and particularly with increasing density, we find that the spiral loses to the 2-parton wavefunction, which we interpret as quasi-1D dominated regime. FIG. 9: Variational phase diagram for the anisotropic 6 × 6 cluster in magnetic field. In this study, we exclude incom- mensurate versions of V and Y states. Due to finite size limi- tation, certain regions are marked as unresolved. In addition, the region δ > 0 . 5 is strongly dominated by quasi-1d physics, particularly for this small cluster study (see discussions in text). visits all phases. On the other hand, for commensurate ⃗ Q = (4 π/ 3 , 0), α = 0 and π/ 2 correspond to distinct V and Ψ-type phases discussed in the isotropic case; both can be viewed as “parent” states for the incommensu- rate coplanar phase, but we will continue referring to the latter as V -type. calculation extended to all J $^{′}$/J . 37 For ease of implementation, we use the same trans- lationally invariant pseudopotentials given in Eq. (22). The V state has an incommensurate density wave and in principle allows more complicated pseudopotentials, so this choice probably biases slightly in favor of the spiral which has uniform boson density. In all other respects, the physical setting and the variational freedom are very similar in our realizations of the spiral and V states, and we think this study provides a fair comparison between the two phases even if the Q$_{x}$ may be slightly off and the Jastrow pseudopotentials are not the most general. Fig. 8 shows the result of our incommensurate V versus spiral variational study. The boundary between the two phases qualitatively agrees with our earlier argument, suggesting that the hard core repulsion is comparatively less important at low density. We note that the obtained trial energies of the V and spiral states are quite close (particularly at low density), hence the exact location of the phase boundary should not be taken as definitive. Furthermore, the simple pseudopotential is clearly not optimal in the highly anisotropic regime, and eventually our spiral loses to the 2-parton states. A more rigorous V versus spiral variational study can be pursued by in- troducing more variational parameters into the Jastrow factor and employing systematic wavefunction optimiza- tion methods. 25 The variational result that the coplanar phase extends to large anisotropy for fields near satu- ration is also in agreement with a recent dilute boson Figure 9 summarizes a variational phase diagram ob- tained for the 36-site cluster considering all boson densi- ties. We label certain parts of the diagram with question marks or broken lines to indicate these regions as unre- solved or less reliable. The figure shows the uud phase extending relatively far into the anisotropic region. On both sides of the uud phase, the commensurate copla- nar phases remain stable over the incommensurate spi- ral for certain ranges of the anisotropy. As the spatial anisotropy biases against the commensurate states, the actual V and Y regions are expected to be wider if the wavefunctions are generalized to incommensurate ver- sions. However, we exclude such extensions since they could not be accommodated on the 6 × 6 cluster. For δ > 0 . 5, our 2-parton trial energies are generally very good. However, we think that this only indicates the onset of quasi-1d physics and strong finite-size effects as discussed earlier for the specific densities. From the present results, we make an interesting ob- servation that at the anisotropy relevant for Cs$_{2}$CuBr$_{4}$, δ ≈ 0 . 3, the transition occurs at density somewhere be- tween n = 1 / 5 (magnetization 0 . 6 of saturation) and n = 1 / 6 (magnetization 2 / 3 of saturation). The V phase occupies the region near saturation, while the spiral oc- curs at lower magnetizations. Thus, if the Heisenberg model is an adequate description, some of the features in the high field phase diagram of Cs$_{2}$CuBr$_{4}$ may be due to the competing umbrella-type and coplanar states. 7 F. Summary of anisotropic study n=1/18 n=1/9 n=1/6 n=1/4 V V V V S S S S S S S S S S S S S S S S 0.0 1.0 / ( J J’ ) J in V in V in V in V in V in V in V in V in V in V in V in V in V in V in V V n=1/5 V V S S S S S in V 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 Spiral V uud Y h 2 3 4 Spiral ? Quasi-1d ? 5 1.0 / ( J J’ ) J ? Fully polarized ?