Parent Hamiltonians for lattice Halperin states from free-boson conformal field theories

We introduce a family of many-body quantum states that describe interacting spin one-half hard-core particles with bosonic or fermionic statistics on arbitrary one- and two-dimensional lattices. The wave functions at lattice filling fraction $\nu=2/(2m+1)$ are derived from deformations of the Wess-Zumino-Witten model $\mathfrak{su}(3)_1$ and are related to the $(m+1,m+1,m)$ Halperin fractional quantum Hall states. We derive long-range SU(2) invariant parent Hamiltonians for these states which in two dimensions are chiral $t$-$J$-$V$ models with additional three-body interaction terms. In one dimension we obtain a generalisation to open chains of a periodic inverse-square $t$-$J$-$V$ model proposed in [Z. N. C. Ha and F. D. M. Haldane, Phys. Rev. B $\mathbf{46}$, 9359 (1992)]. We observe that the gapless low-energy spectrum of this model and its open-boundary generalisation can be described by rapidity sets with the same generalised Pauli exclusion principle. A two-component compactified free boson conformal field theory is identified that has the same central charge and scaling dimensions as the periodic bosonic inverse-square $t$-$J$-$V$ model.