Extended scaling relations for planar lattice models (Q1048139)

The authors prove the validity of some of the exact scaling relations derived in the literature for planar lattice models [\textit{L. P. Kadanoff}, Phys. Rev. Lett. 39, 903--905 (1977); \textit{A. Luther} and \textit{I. Peschel}, Phys. Rev. B 12, 3908--3917 (1975)]. They focus mainly on the Eight Vertex and Ashkin-Teller models, but their result can be extended to several other models. Main result is the following: If the coupling is small enough, the critical indices \(x_{\pm}\) of the Eight Vertex or Ashkin-Teller models verify \[ x_{-}=\frac{1}{x_{+}}, \] \[ \nu=\frac{1}{2-x_{+}}; \] and, in the case of the anisotropic Ashkin-Teller model, \[ x_{T}=\frac{2-x_{+}}{2-x_{-}}. \]