QCD at finite isospin density

QCD at finite isospin chemical potential mu (l) has no fermion sign problem and can be studied on the lattice. We solve this theory analytically in tw o limits: at low mu (l), where chiral perturbation theory is applicable, an d at asymptotically high mu (l), where perturbative QCD works. At low isosp in density the ground state is a pion condensate, whereas at high density i t is a Fermi liquid with Cooper pairing. The pairs carry the same quantum n umbers as the pion. This leads us to conjecture that the transition from ha dron to quark matter is smooth, which passes several tests. Our results imp ly a nontrivial phase diagram in the space of temperature and chemical pote ntials of isospin and baryon number.