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

QCD at a finite isospin chemical potential μI 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 μI, where chiral perturbation theory is applicable, and asymptotically high μI, 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 μI 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 〈ūγ5d〉 and may be the asymptotic limit of the inhomogeneous pion-condensation phase advocated by Migdal and others.