Nonuniversal critical behavior and first-order transitions in a coupledXY-Ising model

We study, by a Migdal-Kadanoff approximation and Monte Carlo simulations, the phase diagram of a two-dimensional coupled XY-Ising model. This model can describe phase transitions in different systems with underlying continuous and ${\mathit{Z}}_2$ symmetries. Depending on the parameters, we find separate XY, Ising and first-order transitions. Also, a line of continuous transitions is found with simultaneous loss of XY and Ising order and varying critical exponents. The fully frustrated XY and Josephson-junction systems can be considered to lie along different paths in the model which can result in nonuniversal behavior if the transition is a single one.