Phase transitions in systems with a coupling to a nonordering parameter

The renormalization-group method is applied to the analysis of phase transitions in systems where the order parameter $s$ is coupled to a nonordering additional variable $y$. A variety of critical and tricritical behaviors or first-order transitions is found as a function of the physical variables and possible macroscopic constraints imposed on the system. For $s^{2}y$ coupling, the correlation function of $y$ was found to be governed by a correlation length which is proportional to that of the order parameter, and by a critical index ${\eta }_y·{\eta }_y=2-2\mathrm{}\alpha$; $\alpha$ here is the specific-heat exponent of the appropriate unconstrained system. The singular part of the susceptibility, ${\chi }_y$, has a critical exponent equal to $\alpha$, the true specific-heat exponent. When the coupling is $s^2{y}^2$, a weaker singularity of ${\chi }_y$ appears. The crossover between this behavior and the one typical to $s^{2}y$ coupling is calculated. $〈y〉$ has a singular part with an exponent $1-\alpha$, in the unconstrained case. The breakdown of the scaling law related to the correlation function of $y$ in the constrained case is discussed.