  We discuss the stability of the antiferromagnetic ground state in two spatial
dimensions. We start with a general analysis, based on Gribov's
current-conservation techniques, of the bosonic modes in systems with magnetic
order. We argue that the Goldstone $\phi$ and Higgs $h$ modes mix in
antiferromagnetic systems, and this leads to an effective $hh\phi$ three-point
interaction. We then analyze the instability of the antiferromagnetic system in
two spatial dimensions by studying the non-perturbative behaviour of the Higgs
boson self-energy using the Dyson--Schwinger equations. The ground state turns
out to be unstable for all values of the three-point coupling. We interpret
this as being due to the formation of a (high-$T_C$) superconducting
condensate. The carrier doping dependence of the energy gap has a general
behaviour that is consistent with high-$T_C$ superconductivity.
Superconductivity co-exists with antiferromagnetic order for large
magnetization, or small doping.
