  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. For this purpose, we study in detail the
simplest non-trivial system beyond the Laughlin states, namely bosons at
filling $\nu=\frac{2}{3}$ 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 $6\hbar$, that produces a unique zero energy solution (as
we have confirmed for up to 12 particles). This state, referred to as the
$\lambda$-state, is not identical to the projected composite-fermion state, but
the following facts suggest that the two might be topologically equivalent: the
two sates have a high overlap; they have the same root partition; the quantum
numbers for their neutral excitations are identical; and the quantum numbers
for the quasiparticle excitations also match. On the quasihole side, we find
that even though the quantum numbers of the lowest energy states agree with the
prediction from the composite-fermion theory, these states are not separated
from the others by a clearly identifiable gap. This prevents us from making a
conclusive claim regarding the topological equivalence of the $\lambda$ state
and the composite-fermion state. Our study illustrates how new candidate states
can be identified from constraining selected many particle configurations and
it would be interesting to pursue their topological classification.
