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, ^', , , , governing the temperature variation at the critical point when x0 become renormalized to the values ₗ=- (1-), ₗ= (1-{^') }, ₗ= (1-), ₗ= (1-),. 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.