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We consider a two dimensional (2D) model of particles interacting in a Landaulevel. We work in a finite disk geometry and take the particles to interactwith a linearly decreasing two-body Haldane pseudo-potential. We show that theground state subspace of this model is spanned by the wave-functions that canbe written as polynomial conformal blocks (of an arbitrary conformal fieldtheory) consistent with the filling fraction (scaling dimension). To removedegeneracies, we then add a quadratic perturbation to the Hamiltonian and showthat; 1. Conformal blocks constructed using the Moore-Read construction (e.g.Laughlin, Pfaffian, and Read-Rezayi states) remain exact eigenstates of thismodel in the thermodynamic limit and 2. By tuning an externally imposedsingle-body $-L_z^2$ potential we can enforce Moore-Read conformal blocks tobecome exact ground states of this model in the thermodynamic limit. We cannotrule out the possibility of residual degeneracies in this limit. This model hasno filling dependence and is comprised only from two-body long-rangeinteractions and external single-body potentials. Our results provide insightinto how conformal block wave-functions can emerge in a Landau level.

Almost linear Haldane pseudopotentials and emergent conformal block wave-functions in a Landau level