Spin-wave singularities: Free energy and equation of state in O(n) spin models near Tc

We give a detailed derivation of the thermodynamics of O(n) spin models, correct to O(ε=4-d), using a generalization of the renormalization-group trajectory integral and noncritical matching technique first introduced by Rudnick and Nelson. The results are valid throughout the critical region, both with and without external magnetic field. We especially emphasize the coexistence-curve behavior for spins with a continuous symmetry (n≥2), deriving detailed expressions for the renormalized spin-wave stiffness (superfluid density) and longitudinal susceptibility. We generalize the parametric representation for the equation of state to include spin-wave singularities, yielding corrections to the so-called ''linear model.'' Our expression for the free energy differs in some details from that given previously by Nicoll and Chang, necessitating minor corrections to previous applications to the weakly interacting Bose gas at constant density