Mean-field approximations for short-range four-body interactions at ?« = 3 5

Abstract

Trial wave functions such as the Moore-Read and Read-Rezayi states, which minimize short-range multibody interactions, are candidate states for describing the fractional quantum Hall effects at filling factors ν = 1 / 2 and 2 / 5 in the second Landau level. These trial wave functions are unique zero-energy states of three-body and four-body interaction Hamiltonians, respectively, but they are not close to the ground states of the Coulomb interaction. Previous studies using extensive parameter scans have found optimal two-body interactions on the sphere that produce states close to these. Here we focus on short-ranged four-body interaction and study two mean-field approximations that reduce the four-body interactions to two-body interactions on the sphere by replacing composite operators with their incompressible ground-state expectation values. We present the results for pseudopotentials of these approximate interactions. A comparison of finite system spectra on the sphere of the four-body and the approximate interactions at filling fraction ν = 3 / 5 shows that these approximations produce good effective descriptions of the low-energy structure of the four-body interaction Hamiltonian. The approach also independently reproduces the optimal two-body interaction inferred from parameter scans. We also show that for n = 3 , but not for n = 4 , the mean-field approximations of the n -body interaction are equivalent to particle-hole symmetrization of the interaction. Within the system sizes accessible, analysis of the spectrum of the mean-field two-body Hamiltonian on the torus was inconclusive, and indicates a competing anisotropic state in the system.