7 field Hamiltonian” for each parton flavor, Here we write the parton hopping amplitudes (which can be complex) as t ( n ) rr ′ = | t ( n ) rr ′ | e ia ( n ) rr ′ ; we also allow site- dependent chemical potentials µ ( n ) r to test CDW tenden- cies. The t ( n ) rr ′ and µ ( n ) r are variational parameters. We solve ˆ H ( n ) mf and fill up the corresponding Fermi seas with N$_{d}$$_{1}$ = N$_{d}$$_{2}$ = N$_{b}$ particles. A valid bosonic wavefunction is obtained by applying a Gutzwiller projection such that every site is either empty ( n$_{b}$ = n$_{d}$$_{1}$ = n$_{d}$$_{2}$ = 0) or con- tains both d$_{1}$ and d$_{2}$ partons ( n$_{b}$ = n$_{d}$$_{1}$ = n$_{d}$$_{2}$ = 1): For each boson configuration, the amplitude is given by a product of two Slater determinants. One feature of this construction follows from the fermionic statistics which provides an inherent repulsive Jastrow effect for the par- ticles. This effect can be tuned as follows, which preserves the “sign structure” of the wavefunction while allowing more variational freedom with parameters p$_{1}$ and p$_{2}$ . Numerical calculations can be performed us- ing efficient determinantal Monte Carlo techniques. 32 Besides treating boson repulsion, we want to have good kinetic energy. We can write the frustrated boson hop- ping amplitudes in Eq. (4) as t ( b ) rr ′ = | t ( b ) rr ′ | e ia ( b ) rr ′ and view this as a problem in an external orbital field producing flux π through each triangle. 33 To capture this in the par- ton treatment, we view d (1) and d (2) as charged particles whose charges add up to that of the boson b ; we therefore require Thus the parton mean field Hamiltonian should contain fluxes such that for the two flavors they add up to the original flux seen by the bosons. We can still make dif- ferent choices, say, for the d$_{1}$ ; however, once the a (1) rr ′ are fixed, then the a (2) rr ′ are uniquely determined. We first discuss what we will call “Chern-Simons” states that realize the idea in Ref. 30. For the d$_{1}$ hopping, we take uniform flux of nπ per triangle, where n is the bo- son density per site. With this choice, the det$_{1}$ Slater de- terminant fills the “lowest Landau level” band and gives a finite lattice version of the usual Chern-Simons factor ∏ $_{i<j}$( z$_{i}$ − z$_{j}$ ). We can loosely view the det$_{1}$ as performing flux attachment transformation from the bosons to the d$_{2}$ fermions. 31 Upon subsequent “flux smearing” mean field, the d$_{2}$ see flux (1 − n ) π per triangle. In the ab- sence of site-dependent chemical potentials and for some rational densities, the det$_{2}$ Slater determinant is gapped, and the boson wavefunction realizes a fractionalized “chi- ral spin liquid”. 30,33,34 We have tried these “topological” states for several densities such as n = 1 / 3 , 1 / 4 , 1 / 6 on the isotropic triangular lattice and found that they are poor compared with the uud and V states described ear- lier. Thus the interesting proposal of plateaus due to chiral spin liquid states is not realized on this lattice. 30 We now specialize to density n = 1 / 3 and allow a chemical potential on the A sublattice: µ$_{A}$ = 0 , µ$_{B}$ = µ$_{C}$ = 0. We find that optimal µ (1) A , µ (2) A are large and pro- duce strong CDW order in the mean field state. When this happens, the trial boson state Eq. (18) is no longer topological in nature. Indeed, if the parton hopping is set to zero, this construction simply gives the classical √$_{3}$ × √$_{3 CDW state. The particles completely occupy}$ sublattice A , and there is a large gap at the parton Fermi level. Adding small hopping does not close this gap but only builds in some charge fluctuations into the parton mean field and thus into the boson trial state. Working perturbatively in t ( n $^{)}$/µ ( n ) A , the leading modification to the classical boson CDW wavefunction is to add config- urations where one particle moves from a site A$_{j}$ to a neighbor r . The amplitude for such a configuration is proportional to t (1) A$_{j}$$_{,r}$t (2) A$_{j}$$_{,r}$/ ( µ (1) A µ (2) A ) ∼ t ( b ) A$_{j}$$_{,r}$/µ ( b ) A , where we have kept track of all signs and introduced schemati- cally boson charge gap µ ( b ) A . The result is similar to the perturbative picture of the CDW working directly in the boson language that motivated the wavefunction Eq. (6). Thus at this level the 2-parton states with strong CDW potential are qualitatively the same as the permanent uud state in Sec. II A. The above leading order structure holds for all 2- parton states satisfying Eq. (19). At higher order, the states will differ, and amplitudes can be complex in general: e.g., the Chern-Simons wavefunction described above is complex-valued. On the other hand, the perma- nent uud wavefunction is real. The boson Hamiltonian Eq. (2) is invariant under complex conjugation in the number basis, and the uud state preserves this symme- try. We can construct a real-valued 2-parton state by taking the d$_{1}$ fluxes to be 0 or π through up or down triangles; the d$_{2}$ partons see correspondingly π and 0 fluxes. We will call this state “ U 1 B ”. It was origi- nally discussed at half-filling in Ref. 35, where (in the absence of chemical potentials) it has Dirac nodes at the Fermi level and realizes so-called Algebraic Spin Liquid state. This particular state has a good trial energy in the Heisenberg model, 24,25,31 and can be viewed as a more elaborate real-valued version of the Laughlin-Kalmeyer state (see Sec. IIC of Ref. 31 for more discussion). Away from half-filling, the U 1 B mean field state has Fermi sur- faces of partons and may be unstable to a mechanism described in Ref. 36. However, this is not a direct con- cern here since we are gapping out the state by adding e ia (1) rr ′ e ia (2) rr ′ = e ia ( b ) rr ′ . (19) 〈{ r$_{k}$ } | ψ$_{2}$$_{p}$ 〉 = det$_{1}$ · det$_{2}$ · | det$_{1}$ | p$_{1}$ − 1 · | det$_{2}$ | p$_{2}$ − 1 , (18) | ψ$_{2}$$_{p}$ 〉 = ˆ P$_{G}$ ∏ q$_{n}$ ∈ FS$_{n}$ d (1) † q$_{1}$ d (2) † q$_{2}$ | 0 〉 . (17) ˆ H ( n ) mf = − ∑ 〈 rr ′ 〉 ( | t ( n ) rr ′ | e ia ( n ) rr ′ d ( n ) † r d ( n ) r ′ + h . c . ) − r µ ( n ) r d ( n ) † r d ( n ) r . (16)