Fisher Renormalization for Logarithmic Corrections

For continuous phase transitions characterized by power-law divergences, Fisher renormalization prescribes how to obtain the critical exponents for a system under constraint from their ideal counterparts. In statistical mechanics, such ideal behaviour at phase transitions is frequently modified by multiplicative logarithmic corrections. Here, Fisher renormalization for the exponents of these logarithms is developed in a general manner. As for the leading exponents, Fisher renormalization at the logarithmic level is seen to be involutory and the renormalized exponents obey the same scaling relations as their ideal analogs. The scheme is tested in lattice animals and the Yang-Lee problem at their upper critical dimensions, where predictions for logarithmic corrections are made. There are both theoretical and practical reasons for sustained interest in thermodynamic systems subject to constraint [1]. Experimental measurements of the critical exponents characterising scaling behaviour at continuous phase transitions may deviate significantly from their ideal theoretical counterparts due to the effects of such constraints. Typically, the theoretical power-law divergence of the specific heat in an ideal system is replaced by a finite cusp in its experimental realization (often called the “real” system). Fisher [2] explained this phenomenon as being due to the effect of hidden variables and established elegant relations between the exponents of the ideal and constrained systems (see also Ref. [3]). Experimentally accessible examples of Fisher renormalization include phase transitions in constrained magnetic and fluid systems (e.g., with fixed levels of impurities) [2], the order-disorder transition in highly compressible ammonium chloride [3, 4], the superfluid λ phase transition present in He-He mixtures in confined films [5], the critical behaviour at nematic-smectic-A transitions in liquid-crystal mixtures [6] and emulsions [7] and possibly the critical behavior observed in random-field Ising systems (such as dilute antiferromagnets in applied fields) [8]. In standard notation, the ideal free energy is written f0(t, h), where t is the reduced temperature and h is the reduced external field. The family S = {α, β, γ, δ, ν, η} of critical exponents characterizes the power-law divergences of the specific-heat, magnetization, susceptibility, correlation length and the correlation function of the ideal system. The four standard scaling relations are (see, e.g., Ref. [9] and references therein) α + dν = 2 , (1) α + 2β + γ = 2 , (2) (δ − 1)β = γ , (3) (2− η)ν = γ , (4) where d represents the dimensionality of the system. For a system under constraint the hidden thermodynamic variable x is conjugate to a force u, such that x(t, h, u) = ∂f(t, h, u) ∂u , (5) where f(t, h, u) represents the free energy of the constrained system. The constraint is written x(t, h, u) = X(t, h, u) , (6) where X(t, h, u) is assumed to be an analytic function. It is further assumed that the free energy of the constrained system can be written in terms of its ideal counterpart f0 as f(t, h, u) = f0(t (t, h, u), h(t, h, u)) + g(t, h, u) (7) where t, h and g are analytic functions of their arguments [2]. The transition is ideal in character if observed at fixed u and the ideal free energy f0(t, h) is recovered when u = 0. Under these circumstances, Fisher established that if the specific-heat exponent for the ideal system α is positive, it is renormalized in the constrained system, together