  We study dense nuclear matter and the chiral phase transition in a SU(2)
parity doublet model at zero temperature. The model is defined by adding the
chiral partner of the nucleon, the N', to the linear sigma model, treating the
mass of the N' as an unknown free parameter. The parity doublet model gives a
reasonable description of the properties of cold nuclear matter, and avoids
unphysical behaviour present in the standard SU(2) linear sigma model. If the
N' is identified as the N'(1535), the parity doublet model shows a first order
phase transition to a chirally restored phase at large densities, $\rho \approx
10 \rho_0$, defining the transition by the degeneracy of the masses of the
nucleon and the N'. If the mass of the N' is chosen to be 1.2 GeV, then the
critical density of the chiral phase transition is lowered to three times
normal nuclear matter density, and for physical values of the pion mass, the
first order transition turns into a smooth crossover.
