QCD at Finite Isospin Density

QCD at finite isospin chemical potential ${\mu }_I$ has no fermion sign problem and can be studied on the lattice. We solve this theory analytically in two limits: at low ${\mu }_I$, where chiral perturbation theory is applicable, and at asymptotically high ${\mu }_I$, where perturbative QCD works. At low isospin density the ground state is a pion condensate, whereas at high density it is a Fermi liquid with Cooper pairing. The pairs carry the same quantum numbers as the pion. This leads us to conjecture that the transition from hadron to quark matter is smooth, which passes several tests. Our results imply a nontrivial phase diagram in the space of temperature and chemical potentials of isospin and baryon number.