QCD at a Finite isospin density: From the pion to quark-antiquark condensation

QCD at a finite isospin chemical potential μ I is studied. This theory has no fermion-sign problem and can be simulated on a lattice by using present-day techniques. We solve this theory analytically in two limits: low μ I , where chiral perturbation theory is applicable, and asymptotically high μ I , where perturbative QCD is at work. At a low isospin density, the ground state is a superfluid pion condensate. At a very high density, it is a Fermi liquid with Cooper pairing. The pairs carry the same quantum numbers as the pions. Motivated by this observation, we put forward a conjecture that the transition from hadron to quark matter is smooth. The conjecture passes several nontrivial tests. Our results imply a nontrivial phase diagram in the space of the temperature and chemical potentials of isospin and baryon number. At asymptotically large values of μ I and small values of the baryon chemical potential, the ground state is in a phase similar to the Fulde-Ferrell-Larkin-Ovchinnikov phase. It is characterized by a spatially modulated superfluid order parameter 〈 ūγ 5 d 〉 and may be the asymptotic limit of the inhomogeneous pion-condensation phase advocated by Migdal and others.