The quantum HMF model: I. Fermions

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 offermions. We consider the Thomas–Fermi approximation that becomes exact in a properthermodynamic limit with a coupling constant k ∼ N. The equilibrium configurations, described by the mean-field Fermi(or waterbag) distribution, are equivalent to polytropes of indexn = 1/2. We show that the homogeneous phase, which is unstable in the classical regime,becomes stable in the quantum regime. The homogeneous phase is stabilized by thePauli exclusion principle. This takes place through a first-order phase transitionwhere the control parameter is the normalized Planck constant. The homogeneousphase is unstable for , metastable for and stable for . The inhomogeneous phase is stable for , metastable for and disappears for (for , there exists an unstable inhomogeneous phase with magnetization ). We point out analogies between the fermionic HMF model and the concept of fermion starsin astrophysics. Finally, as a by-product of our analysis, we obtain new results concerningthe Vlasov dynamical stability of the waterbag distribution which is the ground state of theLynden-Bell distribution in the theory of violent relaxation of the classical HMF model. Weshow that spatially homogeneous waterbag distributions are Vlasov-stable iffϵ ≥ ϵc = 1/3 and spatially inhomogeneous waterbag distributions are Vlasov-stable iffϵ ≤ ϵ* = 0.379 andb ≥ b* = 0.37, whereϵ andb are thenormalized energy and magnetization. The magnetization curve displays a first-order phase transitionat ϵt = 0.352 and the domain of metastability ranges fromϵc toϵ*.