Lattice QCD at finite isospin density at zero and finite temperature

We simulate lattice QCD with dynamical u and d quarks at finite chemical potential, ${\mu }_I,$ for the third component of isospin ${(I}_3)$ at both zero and at finite temperature. At zero temperature there is some ${\mu }_I,$ ${\mu }_c$ say, above which $I_3$ and parity are spontaneously broken by a charged pion condensate. This is in qualitative agreement with the prediction of effective (chiral) Lagrangians which also predict ${\mu }_c{=m}_{\pi }.$ This transition appears to be second order, with scaling properties consistent with the mean-field predictions of such effective Lagrangian models. We have also studied the restoration of $I_3$ symmetry at high temperature for ${\mu }_I>{\mu }_c.$ For ${\mu }_I$ sufficiently large, this finite temperature phase transition is very abrupt, suggesting that it is first order. As ${\mu }_I$ is decreased it appears to become second order connecting continuously with the zero temperature transition.