### Possible contribution of direct exchange to the
superfluidity of $^3 He$

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.

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