The original pion condensation was a liquid crystal like phase in neutron
matter conjectured to occur at high densities. The main neutron coupling
to pions is given by a term like $\vec \sigma \cdot \vec \nabla \Pi$,
where $\vec \sigma$ is the neutron spin operator and the $\Pi$ represents
the pion quantum field. This interaction gives an attraction when
the relative angular momentum of the nucleon and pion is 1, or a p-wave
attraction. If it results in a pion condensate it is sometimes called
a p-wave condensate. Similarly for the s-state with angular momentum 0.
The conjectured liquid crystal phase has a neutron liquid where the spin of the
neutrons tend to be aligned in layers, with opposite spin in opposite
layers as if there were an interaction $\vec \sigma \cdot [\hat z\cos(qz)]$.
That is the pion field with wave vector $q$ has a component that can
be viewed classically, or alternatively that field mode
has a macroscopic occupation number. This is then called a pion condensate.
If this drives a phase transition (rather than a cross over), then the
compressibility of the neutron matter will change dramatically
at the transition.
This change in state
will change things like the mass-radius relation of neutron stars.
Now that
LIGO observations on neutron star mergers etc. can give more
information about
the equation, exotic states like these may be either observed or ruled out.
