Finite temperature transition for 2-flavor lattice QCD at finite isospin density

We simulate 2-flavor lattice QCD at finite isospin chemical potential ${\mu }_I$, for temperatures close to the finite temperature transition from hadronic matter to a quark-gluon plasma. The ${\mu }_I$ dependence of the transition coupling is observed and used to estimate the decrease in the transition temperature with increasing ${\mu }_I$. These simulations are performed on an $8^3\times 4$ lattice at three different quark masses. Our estimate of the magnitude of the fluctuations of the phase of the fermion determinant at small quark-number chemical potential $\mu$, suggest that the position of the small $\mu$ and small ${\mu }_I$ transitions should be the same for ${\mu }_I=2\mu$, and we argue that the nature of these transitions should be the same. For all ${\mu }_I<m_{\pi }$ the smoothness of these transitions and the values of the Binder cumulant $B_4$, indicate that these transitions are mere crossovers, and show no sign of a critical endpoint corresponding to that expected at finite $\mu$. This suggests that this finite $\mu$ critical endpoint, if it exists, lies at $\mu >m_{\pi }/2$ over the considered range of quark masses. For ${\mu }_I>m_{\pi }$ and a small isospin ($I_3$) breaking term $\lambda$, we do find evidence of a critical endpoint which would indicate that, for $\lambda =0$, there is a tricritical point on the phase boundary where the pion condensate evaporates, where this phase transition changes from second to first-order.