The quantum HMF model: II. Bosons

Abstract

We study the thermodynamics of quantum particles with long-range interactions atT = 0. Specifically, we generalize the Hamiltonian mean-field (HMF) model to the case of bosons.We consider the Hartree approximation that becomes exact in a proper thermodynamiclimit with a coupling constant k ∼ 1/N. The equilibrium configurations are solutions of the mean-field Schrödinger equation with acosine interaction. We show that the homogeneous phase, which is unstable in the classicalregime, becomes stable in the quantum regime. The homogeneous phase is stabilized by theHeisenberg uncertainty principle. This takes place through a second-order phase transitionwhere the control parameter is the normalized Planck constant. The homogeneousphase is unstable for and stable for . The inhomogeneous phase is stable for and disappears for . We point out analogies between the bosonic HMF model and the concept of boson starsin astrophysics. We also discuss the differences between bosons and fermions for whatconcerns the thermodynamic limit, the order of the phase transition and the form of thedensity profiles.