Quenched lattice QCD at finite isospin density and related theories

We study quenched QCD at finite chemical potential, $\mu_I$, for the third component of isospin and quenched two-colour QCD at finite chemical potential, $\mu$, for quark number. In contrast to the quenched approximation to QCD at finite quark-number chemical potential, the quenched approximations to these theories behave similarly to the full theories. The reason is that these theories have real positive fermion determinants. In both of these theories there is some critical chemical potential above which the charge coupled to the chemical potential is spontaneously broken. In each case, the transition appears to be second order. For QCD at finite $\mu_I$ and relatively weak gauge coupling ($6/g^2=5.7$) the isospin breaking charged pion condensate appears to scale with mean field critical exponents near the critical point, as expected. For QCD at finite $\mu_I$ and intermediate gauge coupling ($6/g^2=5.5$) and for 2-colour QCD at finite $\mu$, also at intermediate gauge coupling ($4/g^2=2.0$), we again observe a second order transition, but the scaling is not mean field but is instead consistent with the exponents of the tricritical point. This indicates that additional interactions at finite lattice spacing are large enough that there are competing transitions at these larger couplings.