Renormalization of Critical Exponents by Hidden Variables

The thermodynamics of the critical point is discussed generally for a system in which some "hidden variable" $x$, e.g., a density of impurities, fluctuates in equilibrium but is subject to a constraint (e.g., fixed number of impurity atoms). Under rather general assumptions it is shown that the "ideal" critical exponents $\alpha ,{\alpha }^{\prime },\beta ,\gamma ,\Delta ,\mathbf{\cdots }$ governing the temperature variation at the critical point when $x\equiv 0$ become renormalized to the values ${\alpha }_X=\frac{-\alpha }{(1-\alpha )},{\beta }_X=\frac{\beta }{(1-{\alpha }^{\prime })},{\gamma }_X=\frac{\gamma }{(1-\alpha )},{\Delta }_X=\frac{\Delta }{(1-\alpha )},\mathbf{\cdots }$. A variety of exactly soluble models are discussed, including a "mobile-electron Ising ferromagnet," for which this renormalization can be checked explicitly. The relevance of these results for the interpretation of theoretical and experimentally observed critical exponents is considered briefly.