  We study the effective interactions between Dirac fermions on the surface of
a three-dimensional topological insulator due to the proximity coupling to the
magnetic fluctuations in a ferromagnetic or antiferromagnetic insulator. Our
results show that the magnetic fluctuations can mediate attractive interactions
between Dirac fermions of both Amperean and BCS type. In the ferromagnetic
case, we find pairing between fermions with parallel momenta, so-called
Amperean pairing, whenever the effective Lagrangian for the magnetic
fluctuations does not contain a quadratic term. The pairing interaction also
increases with increasing Fermi momentum, and is in agreement with previous
studies in the limit of high chemical potential. If a quadratic term is
present, the pairing is instead of BCS type above a certain chemical potential.
In the antiferromagnetic case, BCS pairing occurs when the ferromagnetic
coupling between magnons on the same sublattice exceeds the antiferromagnetic
coupling between magnons on different sublattices. Outside this region in
parameter space, we again find that Amperean pairing is realized.
