Fluctuation induced critical behavior at nonzero temperature and chemical potential

We discuss phase transitions in relativistic systems as a function of both the chemical potential and temperature. The presence of a chemical potential explicitly breaks Lorentz invariance and may additionally break other internal symmetries. This introduces new subtleties in the determination of the critical properties. We discuss separately three characteristic effects of a nonzero chemical potential. First, we consider only the explicit breaking of Lorentz invariance using a scalar field theory with a global $U\left(1\right)\mathrm{}$ symmetry. Second, we study the explicit breaking of an internal symmetry in addition to Lorentz invariance using two-color QCD at nonzero baryonic chemical potential. Finally, we consider the spontaneous breaking of a symmetry using three-color QCD at nonzero baryonic and isospin chemical potential. For each case, we derive the appropriate three-dimensional effective theory at criticality and study the effect of the chemical potential on the fixed point structure of the $\beta$ functions. We find that the order of the phase transition is not affected by the explicit breaking of Lorentz invariance but is sensitive to the breaking of additional symmetries by the chemical potential.