The quantum HMF model: I. Fermions

We study the thermodynamics of quantum particles with long-range interactions at T = 0. Specifically, we generalize the Hamiltonian mean-field (HMF) model to the case of fermions. We consider the Thomas-Fermi approximation that becomes exact in a proper thermodynamic limit N -> +infinity with a coupling constant k similar to N. The equilibrium configurations, described by the mean-field Fermi (or waterbag) distribution, are equivalent to polytropes of index n = 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 the Pauli exclusion principle. This takes place through a first-order phase transition where the control parameter is the normalized Planck constant. The homogeneous phase is unstable for (h) over bar < <(h)over bar>(c) = 2/root pi, metastable for (h) over bar (c) < <(h)over bar> < <(h)over bar>(t) equivalent to 1.16 and stable for (h) over bar > (h) over bar (t). The inhomogeneous phase is stable for (h) over bar < <(h)over bar>(t), metastable for (h) over bar (t) < <(h)over bar> < <(h)over bar>(*) equivalent to 1.18 and disappears for (h) over bar > (h) over bar (*) (for (h) over bar (c) < <(h)over bar> < <(h)over bar>(*), there exists an unstable inhomogeneous phase with magnetization 0 < b < b(*) equivalent to 0.37). We point out analogies between the fermionic HMF model and the concept of fermion stars in astrophysics. Finally, as a by-product of our analysis, we obtain new results concerning the Vlasov dynamical stability of the waterbag distribution which is the ground state of the Lynden-Bell distribution in the theory of violent relaxation of the classical HMF model. We show that spatially homogeneous waterbag distributions are Vlasov-stable iff epsilon >= epsilon(c) = 1/3 and spatially inhomogeneous waterbag distributions are Vlasov-stable iff epsilon <= epsilon(*) = 0.379 and b >= b(*) = 0.37, where epsilon and b are the normalized energy and magnetization. The magnetization curve displays a first-order phase transition at epsilon(t) = 0.352 and the domain of metastability ranges from epsilon(c) to epsilon(*).