Search for exact local Hamiltonians for general fractional quantum Hall states

We report on our systematic attempts at finding local interactions for which the lowest-Landau-level projected composite-fermion wave functions are the unique zero energy ground states. We study in detail the simplest non-trivial system beyond the Laughlin states, namely bosons at filling factor 2/3 and identify local constraints among clusters of particles in the ground state. By explicit calculation, we show that no Hamiltonian up to (and including) four particle interactions produces this state as the exact ground state, and speculate that this remains true even when interaction terms involving greater number of particles are included. Surprisingly, we can identify an interaction, which imposes an energetic penalty for a specific entangled configuration of four particles with relative angular momentum of 6, that produces a unique zero energy solution (as we have confirmed for up to 12 particles). This state is not identical to the projected CF state, but have high overlaps with the CF state and the same root partition, quasiparticle and neutral excitation spectrum as the CF state. On the quasihole side, the quantum numbers of the low energy states agree with the CF state but these states are not separated from the others by a clearly identifiable gap.