Truncated Calogero-Sutherland models

A one-dimensional quantum many-body system consisting of particles confined in a harmonic potential and subject to finite-range two-body and three-body inverse-square interactions is introduced. The range of the interactions is set by truncation beyond a number of neighbors and can be tuned to interpolate between the Calogero-Sutherland model and a system with interactions among nearest and next-nearest neighbors discussed by Jain and Khare. The model also includes the Tonks-Girardeau gas describing impenetrable bosons as well as a novel extension with truncated interactions. All these systems are exactly solvable and exhibit a linear spectrum, with the effect of the interactions being absorbed in a nontrivial zero-point energy. We characterize the degeneracies and derive the canonical partition function. While the ground state wavefunction takes a truncated Bijl-Jastrow form, excited states are found in terms of multivariable symmetric polynomials. We numerically compute the density profile and one-body reduced density matrix of the ground state and discuss the effect of the strength and finite range of the interaction potential.