  While pure Yang-Mills theory feature the centre symmetry, this symmetry is
explicitly broken by the presence of dynamical matter. We study the impact of
the centre symmetry in such QCD-like theories. In the analytically solvable
Schwinger model, centre transitions take place even under extreme conditions,
temperature and/or density, and we show that they are key to the solution of
the Silver-Blaze problem. We then develop an effective SU(3) quark model which
confines quarks by virtue of centre sector transitions. The phase diagram by
confinement is obtained as a function of the temperature and the chemical
potential. We show that at low temperatures and intermediate values for the
chemical potential the centre dressed quarks undergo condensation due to Bose
like statistics. This is the Fermi Einstein condensation. To corroborate the
existence of centre sector transitions in gauge theories with matter, we study
(at vanishing chemical potential) the interface tension in the
three-dimensional Z2 gauge theory with Ising matter, the distribution of the
Polyakov line in the four-dimensional SU(2)-Higgs model and devise a new type
of order parameter which is designed to detect centre sector transitions. Our
analytical and numerical findings lead us to conjecture a new state of cold,
but dense matter in the hadronic phase for which Fermi Einstein condensation is
realised.
