13 FIG. 10: Schematic phase diagram for the anisotropic trian- gular lattice in magnetic field, combining 6 × 6 study as well as larger cluster studies. We suggest the possibility that the uud plateau extends across the entire range of anisotropy. The V phase is commensurate (see Fig. 2) near the isotropic axis and the plateau but becomes incommensurate at moderate anisotropy and higher fields. The highly anisotropic region is not well resolved in our variational study. The low field regime is also not studied thoroughly in this work, while pre- vious studies 20,23,25 suggest spin liquid state along h = 0 and δ > 0 . 2. Figure 10 shows a schematic phase diagram based on the 6 × 6 anisotropic study as well as studies on larger clusters. Here, we address a number of unresolved re- gions in Fig. 9: limits of the uud plateau, the boundary between V and spiral, and the boundary between com- mensurate and incommensurate V . We find that the uud phase extends much futher and may be even to all δ . Also, a significant portion of the phase diagram at high fields is occupied by the incommensurate V phase. We note that our work does not rule out other in- commensurate phases found in the recent study by Al- icea. et al 9 For example, we were not able to come up with a good implementation of the incommensurate ex- tension of the Y state. On the other hand, we did try Bijl- Jastrow-type wavefunctions for distorted umbrella states discussed in Ref. 9, which are commensurate supersolids with incommensurate spiral phase angles. On the 36-site cluster, these trial states optimized to the incommensu- rate spiral with uniform boson density, but we have not explored this thoroughly on larger clusters. Overall, our results are more conclusive at low densities and much less at high densities between 1/3 and 1/2. We studied the Heisenberg antiferromagnet on the spatially anisotropic triangular lattice in the field from a variational perspective. On the isotropic lattice, we IV. SUMMARY AND DISCUSSION constructed a very simple and physically transparent permanent-type wavefunction for the uud state at density 1 / 3. This is a Mott insulator of bosons where we accu- rately included small charge fluctuations by using appro- priate localized boson orbitals. The remarkable trial en- ergy suggests that such approach may be useful in other Mott insulator contexts. Next, we obtained natural ex- tensions to nearby V and Y supersolid phases respec- tively for n < ∼ 1 / 3 and n > ∼ 1 / 3, where the physics re- mains strongly influence by the proximity to n = 1 / 3. By connecting to a Bijl-Jastrow-type candidate wave- function at low density, the coplanar V phase extends to all n < 1 / 3 (i.e., up to the saturation field in the spin model language). Note, however, that at very low density another coplanar state (Ψ-type) is expected to be very close, 11 and we cannot resolve between the two. On the higher density side of the plateau (i.e., at lower fields), the permanent-type Y wavefunction performs well near the plateau but narrowly loses to the Huse-Elser spiral candidate at densities close to half-filling (zero field). The latter result is consistent with other recent works. 21,24,26 The success of our isotropic study encouraged us to extend it to the anisotropic lattice. At density n = 1 / 3, we begin with the permanent-type realization of the uud and then connect to a conceptually similar but techni- cally different 2-parton realization at higher anisotropy. Surprisingly, we found that the uud phase remains stable over a large range of anisotropy. In conjunction with the same CDW phase found in the decoupled chains limit, 10 we suggest that the uud phase may in fact extend across the entire range of anisotropy. This conjecture can be tested more rigorously using a DMRG study on finite- width strips. In the low boson density region (i.e., at high fields), the Bijl-Jastrow-type V commensurate supersolid wave- function is smoothly connected to the incommensurate version. This state competes with the incommensurate spiral, and we can accurately compare the two. We found that the incommensurate V state has lower energy in a large region of the phase diagram, extending up to a fairly large value of anisotropy in the very dilute regime (i.e., close to the saturation fields). On the other hand, the V phase remains commensurate near the isotropic axis and the plateau. In the high density regime (i.e., at low fields), we at- tempted to construct an incommensurate Y candidate using a Bijl-Jastrow-type wavefunction but found that this construction performs poorly. This low field region at moderate to high lattice anisotropy calls for more com- prehensive investigation. One of the goals we had was to explore possible new plateaus in the high field regime of Cs$_{2}$CuBr$_{4}$. We have learned that the phase diagram is already very rich even without considering any additional plateaus. Neverthe- less, for several densities such as 1 / 6, 2 / 9, and 1 / 4 we implemented permanent-type wavefunctions for various proposed CDW from Ref. 7 as well as for some addi- tional stripe-like orderings, and inevitably found that ei- 1 0 0.2 0.4 0.6 0.8 Spiral V uud Y h 2 3 4 Spiral 5 1.0 / ( J J’ ) J Fully polarized ? I$_{nc}$$_{om}$$_{m}$ e$_{ns}$$_{ur}$ a t e Q$_{uas}$ i-$_{1d}$ ? ? ? Quasi-1d ? ? ? V C$_{om}$ m e$_{ns}$$_{ur}$ a te ? ? Spin-liquid ?