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Color superconductivity is a phenomenon predicted to occur in quark matter if the baryon density is sufficiently high and the temperature is not too high. Color superconducting phases are to be contrasted with the normal phase of quark matter, which is just a weakly interacting Fermi liquid of quarks.
In theoretical terms, a color superconducting phase is a state in which the quarks near the Fermi surface become correlated in Cooper pairs, which condense. In phenomenological terms, a color superconducting phase breaks some of the symmetries of the underlying theory, and has a very different spectrum of excitations and very different transport properties from the normal phase.
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