Crossover functions by renormalization-group matching: O(ε²) results

Abstract

By considering the relationship of the matching techniques of Bruce and Wallace to the differential renormalization-group generators, we find that a restatement of the former gives improved results with the same number of perturbative terms. In particular, the vertex functions and specific heat of a n-component spin system are given exactly in the spherical limit n→∞ even at first order in perturbation theory (T>T_c). The nature of the nonlinear scaling variables is clarified, and the results are generally expressed in a more compact form. The general n-component disordered phase functions are rederived to O(ε²), where ε=4-d. The cross-over equations for the n=1 Ising-like case are derived for the Helmholtz potential A(M), the magnetic field h/M, the inverse susceptibility Γ₂, and the correlation length ξ to O(ε²).