2 TABLE I: Comparison of uud trial energies for different num- ber of short-range Jastrow parameters for 12 bosons on a 6 × 6 cluster. Localized orbitals in the permanent extend only to nearest neighbour sites with amplitude α , which is single vari- ational parameter in the first listed uud case. The second uud case has one nearest-neighbor (nnb) Jastrow pseudopotential which is taken to be the same between any pair of nnb sites, while the third case has two such parameters, one for A − B and A − C nnb pairs and the other for B − C nnb pairs, as is appropriate given lattice symmetries of the uud state. We also show trial energy for the spiral state with 4 variational parameters (same as in table III); this state performs poorly compared to the uud state. FIG. 3: Boson interpretation of the spin orderings in Fig. 2. Gray scale shows density order 〈 n$_{r}$ 〉 , while arrows show su- perfluid order 〈 b † r 〉 . a) Spiral is a uniform superfluid phase with rotating phase angles; c) uud is a Mott insulator with √$_{3}$ × √$_{3 CDW order; b,d)}$ Y and V are supersolid phases which contain both charge density and particular superfluid orderings. (a) (c) (b) (d) n=0 n=1 Boson density Trial state N orb par N Jas short N Jas long N tot par E/bond classical -0.0833 spiral 0 1 2 4 -0.1265 uud 1 0 0 1 -0.1347 1 1 0 2 -0.1354 1 2 0 3 -0.1355 exact -0.1361 II. ISOTROPIC TRIANGULAR ANTIFERROMAGNET: 6 × 6 STUDY variational study. While these are simple to draw, re- alizing them as wavefunctions is non-trivial. The spiral phase is a boson superfluid containing rotating phase an- gles (see Fig. 3a) and is captured by an elegant Huse-Elser generalization of the Bijl-Jastrow wavefunction. 22 For the other phases, constructing simple and yet accurate wave- functions is not as straightforward and requires consider- ation of their physical nature in terms of bosons. Thus, the uud is a Mott insulating phase (see Fig. 3c) which re- quires a wavefunction with strongly localized bosons and rapidly decaying correlations. The Y phase is an interest- ing supersolid phase (see Fig. 3b) with rapidly decaying boson correlations between sites on one of the three sub- lattices as well as long-range correlation between sites on the other two sublattices. The V phase is a different su- persolid (see Fig. 3d) with long-range boson correlations between all sites; here we find that two different construc- tions of the trial wavefunctions are required to capture the lower and higher density regimes. (We remark that supersolid phases of bosons on the half-filled triangular lattice and in the presence of strong repulsion have been of much recent interest, see Refs. 21,24,26 and citations therein.) In section II, we present simple, few-parameter can- didate wavefunctions used in our isotropic study. En- couraged by the accuracy of these candidates, we gener- alize the wavefunctions to incommensurate versions for our anisotropic study in section III. We conclude with a discussion of the results and implications for Cs$_{2}$CuBr$_{4}$ in section IV. At density n = 1 / 3, the uud phase is stabilized by quantum fluctuations. We construct a simple bo- son wavefunction by using N$_{b}$ = N/ 3 orbitals localized with simple choices of pseudopotentials u$_{rr}$ ′ providing additional variational freedom. N$_{b}$ ≥ 13 data is taken from Bernu et al . We perform all studies at fixed boson number N$_{b}$ . For each wavefunction below, we also include a Jastrow factor In this section, we consider the isotropic triangular Heisenberg antiferromagnet with J$_{rr}$ ′ = J for all nearest neighbour links. Beginning at density n = 1 / 3, where we have an excellent wavefunction for the uud Mott in- sulator phase of Fig. 3c, we construct similarly inspired wavefunctions for nearby supersolid phases of Figs. 3b,d. Next, we describe Bijl-Jastrow-type wavefunctions for the spiral of Fig. 3a and V supersolid of Fig. 3d, which perform better for n further from 1/3. We also discuss an alternative construction of the uud state using a det × det (“2-parton”) trial wavefunction. For the ED calculations, we compute ground state energies for N$_{b}$ ≤ 12, while 13 Jastrow( { n$_{r}$ } ) = e − 1 2 ∑$_{r,r}$$_{′}$ u$_{rr}$ ′ n$_{r}$n$_{r}$ ′ (5) A. uud state at n = 1 / 3