  We describe a novel superconducting phase that arises due to a pairing
instability of the half-metallic antiferromagnetic (HM AFM) normal state. This
single spin superconducting (SSS) phase contains broken time reversal symmetry
in addition to broken gauge symmetry, the former due to the underlying magnetic
order in the normal state. A classification of normal state symmetries leads to
the conclusion that the HM AFM normal phase whose point group contains the
inversion operator contains the least symmetry possible which still allows for
a zero momentum pairing instability. The Ginzburg-Landau free energy for the
superconducting order parameter is constructed consistent with the symmetry of
the normal phase, electromagnetic gauge invariance and the crystallographic
point group symmetry including inversion. For cubic, hexagonal and tetragonal
point groups, the possible symmetries of the superconducting phase are
classified, and the free energy is used to construct a generalized phase
diagram. We identify the leading candidate out of the possible SSS phases for
each point group. The symmetry of the superconducting phase is used to
determine the cases where the gap function has generic zeros (point or line
nodes) on the Fermi surface. Such nodes always occur, hence thermodynamic
properties will have power-law behavior at low temperature.
