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Journal of Physics A Mathematical and General

The ground state of chargeless fermions with finite magnetic moment

S. D. Mahanti, Sudhanshu Shekhar Jha

Published Jan 24, 2006• 5 citations
Publisher

Abstract

We consider the ground state of a system of chargeless fermions, such as neutrinos, of mass m and magnetic moment p interacting through long-range magnetic dipole interaction, within the framework of a Hartree-Fock variational approach. At high densities the uniform paramagnetic state becomes unstable towards a ferromagnetic state with quadrupolar deformation of the Fermi surface. The exchange energy which is attractive dominates the repulsive kinetic energy. If we let the density be a variable, then above a certain density the system will collapse to an infinite density state unless another short-range interaction stops the collapse. In the case of large deformations, the possibility of a purely dipolar deformation exists.

DOI
10.1088/0305-4470/39/6/002

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We consider the ground state of a system of chargeless fermions, such as neutrinos, of mass m and magnetic moment p interacting through long-range magnetic dipole interaction, within the framework of a Hartree-Fock variational approach. At high densities the uniform paramagnetic state becomes unstable towards a ferromagnetic state with quadrupolar deformation of the Fermi surface. The exchange energy which is attractive dominates the repulsive kinetic energy. If we let the density be a variable, then above a certain density the system will collapse to an infinite density state unless another short-range interaction stops the collapse. In the case of large deformations, the possibility of a purely dipolar deformation exists.
We obtain the best upper bound for the ground-state energy of a system of chargeless fermions of mass m, spin s=1/2 , and magnetic moment mus[over ] as a function of its density in the fully spin-polarized Hartree-Fock determinantal state, specified by a prolate spheroidal plane-wave single-particle occupation function n_(k[over ]) , by minimizing the total energy E at each density with respect to the variational spheroidal deformation parameter beta(2),0< or =beta(2)< or =1 . We find that at high densities, this spheroidal ferromagnetic state is the most likely ground state of the system, but it is still unstable towards the infinite-density collapse. This optimized ferromagnetic state is shown to be a stable ground state of the dipolar system at high densities, if one has an additional repulsive short-range hardcore interaction of sufficient strength and nonvanishing range.
We show that the ground state of a system of magnetic dipoles, with no electric charge, is a ferromagnetic quantum Fermi liquid at high densities, driven by the dipolar exchange energy. As in the system of classical point dipoles, the direct dipole energy is zero in this case. With decreasing density, there is a transition to an antiferromagnetic lattice state. An addition of short range hard core repulsive potential will arrest the infinite density collapse of the ferromagnetic state, and possible melting of the low density antiferromagnetic lattice state.
The stability condition of the Hartree-Fock states, which are constructed out of the plane waves times spin-up or -down eigenfunctions. i.e. the paramagnetic (P), ferromagnetic (F) and unsaturated ferromagnetic (UF) states, is examined in some detail in the case of repulsive δ-function. Yukawa and Coulomb interactions. The spherical Fermi surface leads of course to the minimum energy in each case; this energy is then minimized as the function of polarization n=(n+-n-)/(n++n-,n± being the total number of spin-up and -down electrons. We thus can determine the domains of density where each of the three states is most stable. Then the stability of these states against small deformations of one-particle wave functions is examined in their respective domains. In the case of δ-function potential no instability occurs, while for an electron gas the P state (rs ≪ 5.45) is always unstable and the F state (rs > 5.45) is unstable near the critical density. The variational method in the form of canonical transformation is presented to obtain a new Hartree-Fock solution having helical spin structure in case instability occurs. The case of δ-function interaction is worked out for the P state and it is actually shown that such a solution can exist if and only if the P state becomes unstable. This new solution is of non-perturbative nature in the sense that it can never be obtained by perturbation theory starting from the free ground state.
We study in this paper the effect of weak, short-ranged interaction on disordered metals. Through analyzing the interaction matrix elements between different eigenstates of the noninteracting and corresponding Hartree-Fock single-particle Hamiltonian, we argue that as a result of localized single-particle eigenstates around the Fermi surface, the quasiparticle states on the Fermi surface are unstable toward formation of magnetic moments for arbitrary weak (but finite) repulsive interaction in the thermodynamic limit. This is a mechanism very different from the case of strong interaction $U\ensuremath{\sim}{W}_{B}$ (WB={W}_{B}=WB​= bandwidth) or the quantum Griffiths effect where local moments are formed at small localized regions where coupling to the surrounding is weak. Numerical simulations are performed to verify our analysis. We further propose within a Landau Fermi-liquid-type framework that our result is applicable for general electronic systems with weak, short-ranged interaction as long as the quasiparticle states exist and are localized. An analogous result is obtained for attractive interaction, suggesting that Fermi glass state is intrinsically unstable in arbitrary dimension.
The condition that the plane wave state is the lowest Hartree-Fock state of a large fermion system with repulsive interactions, is generally investigated and applied to the systems with $\ensuremath{\delta}$-type interactions, finite-range interactions, and long-range Coulomb interactions. It is found that for finite-range interactions the plane wave state is stable in both the high-density and low-density regions, but may become unstable in the intermediate-density region. The high-density stable region disappears for $\ensuremath{\delta}$-type interactions, while the low-density stable region vanishes for long-range Coulomb interactions. Especially for an electron gas, the critical value rs{r}_{s}rs​ is smaller than 4.5. A speculation concerning the transition of the true ground state is discussed.