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Trial wavefunctions like the Moore-Read and Read-Rezayi states which minimizeshort range multibody interactions are candidate states for describing thefractional quantum Hall effects at filling factors $\nu = 1/2$ and $\nu = 2/5$in the second Landau level. These trial wavefunctions are unique zero energystates of three body and four body interaction Hamiltonians respectively butare not close to the ground states of the Coulomb interaction. Previous studiesusing extensive parameter scans have found optimal two body interactions thatproduce states close to these. Here we focus on short ranged four bodyinteraction and study two mean field approximations that reduce the four bodyinteractions to two body interactions by replacing composite operators withtheir incompressible ground state expectation values. We present the resultsfor pseudopotentials of these approximate interactions. Comparison of finitesystem spectra of the four body and the approximate interactions at fillingfraction $\nu=3/5$ show that these approximations produce good effectivedescriptions of the low energy structure of the four body ineractionHamiltonian. The approach also independently reproduces the optimal two bodyinteraction inferred from parameter scans. We also show that for $n=3$, but notfor $n=4$, the mean field approximations of the $n$-body interaction isequivalent to particle hole symmetrization of the interaction.

Mean field approximations for short range four body interactions at $\nu=3/5$