  We present a detailed discussion of the thermodynamics of the parity-doublet
nucleon-meson model within a mean-field theory, at finite temperature and
baryon-chemical potential, with special emphasis on the chiral transition at
large baryon densities and vanishing temperature. We consider isospin-symmetric
matter. We systematically compare the parity-doublet model to a related singlet
model obtained by disregarding the chiral partner of the nucleon. After
studying the ground state properties of nuclear matter, the nuclear liquid-gas
transition, and the density modifications of the nucleon sigma term which
govern the low-density regime, we give new insight into the underlying
mechanisms of the zero-temperature chiral transition occurring at several times
the nuclear saturation density. We show that the chiral transition is driven by
a kind of symmetry energy that tends to equilibrate the populations of opposite
parity baryons. This symmetry energy dictates the composition of matter at
large baryon densities, once the phase space for the appearance of the
negative-parity partner is opened. We furthermore highlight the characteristic
role, within the thermodynamics, of the chiral-invariant mass of the
parity-doublet model. We include the chiral limit into all of our discussions
in order to provide a complete picture of the chiral transition.
