Hartree-Fock variational bounds for ground state energy of chargeless fermions with finite magnetic moment in the presence of a hard core potential: A stable ferromagnetic state

We use different determinantal Hartree-Fock (HF) wave functions to calculate true variational upper bounds for the ground state energy of N spin-half fermions in volume V0, with mass m, electric charge zero, and magnetic moment µ, interacting through magnetic dipole-dipole interaction. We find that at high densities when the average interparticle distance r0 becomes small compared to the magnetic length rm ≡ 2mµ2/ħ2, a ferromagnetic state with spheroidal occupation function n↑(\( \vec k \)), involving quadrupolar deformation, gives a lower upper bound compared to the variational energy for the uniform paramagnetic state or for the state with dipolar deformation. This system is unstable towards infinite density collapse, but we show explicitly that a suitable short-range repulsive (hard core) interaction of strength U0 and range a can stop this collapse. The existence of a stable equilibrium high density ferromagnetic state with spheroidal occupation function is possible as long as the ratio of coupling constants Γcm ≡ (U0a3/µ2) is not very small compared to 1.