  We consider a two dimensional (2D) model of particles interacting in a Landau
level. We work in a finite disk geometry and take the particles to interact
with a linearly decreasing two-body Haldane pseudo-potential. We show that the
ground state subspace of this model is spanned by the wave-functions that can
be written as polynomial conformal blocks (of an arbitrary conformal field
theory) consistent with the filling fraction (scaling dimension). To remove
degeneracies, we then add a quadratic perturbation to the Hamiltonian and show
that; 1. Conformal blocks constructed using the Moore-Read construction (e.g.
Laughlin, Pfaffian, and Read-Rezayi states) remain exact eigenstates of this
model in the thermodynamic limit and 2. By tuning an externally imposed
single-body $-L_z^2$ potential we can enforce Moore-Read conformal blocks to
become exact ground states of this model in the thermodynamic limit. We cannot
rule out the possibility of residual degeneracies in this limit. This model has
no filling dependence and is comprised only from two-body long-range
interactions and external single-body potentials. Our results provide insight
into how conformal block wave-functions can emerge in a Landau level.

Strongly Correlated Electrons;High Energy Physics - Theory;Quantum Physics