A Differential Generator for the Free Energy and the Magnetization Equation of State~

The first renormalization group methods for the calculation of the equation of state utilized entirely field theoretic techniques [1]. Recently, equations of state have been obtained by using a combination of field theoretic and differential renormalization group methods [2, 3]. In this work, we introduce a calculational method which is entirely differential in character. This differential generator for the free energy thus avoids the use of any field theoretic diagram expansions and only convergent integrals are needed. The differential generator is derived in a manner analogous to that of ref. [4], except that a saddle point expansion is employed in order to obtain a one-particle irreducible generator [1]. Making the approximation that the expansion coefficients are momentum independent [5], we obtain aA(M, 1)/Zil = exp(—dl)tr ln [~, + exp(21) a2A /3M’~JMJ] (1)