Exact differential real-space renormalization: Ising, Gaussian and Ashkin-Teller models

Two triangular Ising models are coupled by a small four-spin interaction of a generalized Ashkin-Teller type. All interactions are spatially dependent. To lowest order in perturbation theory a closed system of exact real-space renormalization equations is derived. From this system a set of nine partial differential equations decouples which describes the renormalization of nine “collective” variables: three pair interactions and six four-spin interactions. The study of these equations has revealed a marginal direction, which we interpret as a line of fixed points. Its relation to Baxter's line is discussed. Our study of the Ashkin-Teller model is preceded by a coherent presentation of the differential renormalization equations for the Ising and Gaussian model.