Phys. Rev. B 55, 8997 (1997).
The properties of normal and superfluid $^3 He$ have been described using a Hubbard model. We point out that in such a description it is natural to include a contribution from direct exchange, arising from an off-diagonal matrix element of the $^3 He-^3He$ interaction in a local orbital representation. We discuss a new derivation of an effective Hamiltonian for $^3He$, which results in a lattice model with on-site repulsive interaction $U$, nearest-neighbor attraction $V$, and nearest-neighbor ferromagnetic exchange $J$. We examine the stability of p-wave and d-wave superfluid states as a function of the parameters in the Hamiltonian and the particle density, as well as the competition between superfluid and ferromagnetic states within this model. The possible relevance of these results to the physics of $^3He$ is discussed.