Marginality, universality, and expansion techniques for critical lines in two dimensions

Many $d=2\mathrm{}$ phase transition problems have lines of critical points on which critical behavior varies continuously as a function of a single parameter $\lambda$. In treating these problems, it is important to know how $\lambda$ depends upon the other parameters in the Hamiltonian. This paper is concerned with the developments and application of techniques for getting perturbation expansions of $\lambda$ in other system parameters. Two examples are described in detail: the Gaussian model and its near relative the low-temperature planar model. The Gaussian model serves as a kind of base problem in which the effects of marginality may be fully explored. We show explicitly how the planar model may be mapped into a Gaussian model and extend the Kosterlitz-Thouless analysis to calculate the marginal parameter to fourth order in $y_0$.