  We show that even in the limit of large cyclotron gap, Landau level (LL)
mixing can be dominant with scale-free interaction between charged ``elementary
particles" in a fractional quantum Hall system, as long as the filling factor
exceeds certain critical values. The corresponding Hamiltonians with scale-free
interaction can serve as exact model Hamiltonians for certain composite Fermion
or parton states (unlike the well-known TK Hamiltonians where the number of LLs
needs to be fixed by hand), and they are natural physical Hamiltonians for 2D
systems embedded in higher-dimensional space-time. Even with LL mixing the null
spaces of such Hamiltonians (spanned by the ground state and the quasiholes)
can be analytically obtained, and we show these are the generalisation of the
conformal Hilbert spaces (CHS) to more than one LLs. The effective interaction
between the anyons for these topological phases emerges from the kinetic energy
of the ``elementary particles", leading to an interesting duality between
strongly and weakly interacting systems that can be easily understood via the
tuning of parameters in the scale-free interaction. We also propose a novel
experimental platform for approximately realising such model Hamiltonians with
trion like particles, that can potentially lead to very robust (non-Abelian)
FQH phases from two-body Coulomb-based interaction.
