  We present a lattice field theory of spins coupled to Dirac fermions, as a
model for the doped copper oxide compounds. Both the fermionic and spin degrees
of freedom are treated dynamically. The influence of the charge carriers on the
magnetic ordering follows automatically. The magnetic phase diagram at zero
temperature is studied numerically and with Mean-Field methods. The Hybrid
Monte Carlo algorithm is adapted to the O(3) non-linear sigma model constraint.
The charged excitations in the various phases are studied at the Mean-Field
level. Bound states of two charged fermions are found in a strongly coupled
paramagnetic phase, without requiring a Fermi sea. We acquire a qualitative
understanding of high-T$_c$ superconductivity through Bose-Einstein
condensation of those dynamically bound pairs. The model also implies
insulating behaviour at low doping, and Fermi liquid behaviour at large doping.
We predict the possibility of reentrant superconductivity and the absence of
superconductivity in spin-1 and spin-3/2 materials.
