Parent Hamiltonian Construction of Generalized Calogero-Sutherland Models

Abstract: The Calogero-Sutherland model is a paradigmatic integrable system describing one-dimensional non-relativistic particles with inverse-square interactions. At interaction strength $λ=2$, the CSM exhibits a deep connection to anyon physics, featuring the Laughlin-Jastrow polynomial as its exact ground state. Motivated by this structure, we develop a general reverse-engineering construction of positive semi-definite continuum parent Hamiltonians for trial states admitting a rational conformal field theory description with central charge $c<1$. By leveraging the null-vector structure of the underlying primary fields and the associated Belavin-Polyakov-Zamolodchikov equations, we derive corresponding many-body annihilation operators. We then apply this construction explicitly to the Moore-Read and $k=3$ Read-Rezayi states - relating to Ising and Fibonacci anyons, respectively - obtaining continuum Hamiltonians for which these Jack-polynomial states are exact zero modes. We emphasize, however, that our construction does not by itself establish ground-state uniqueness or determine the nature of the excitation spectrum.