Derivation of extended scaling relations between critical exponents in two-dimensional models from the one-dimensional Luttinger model

The extended scaling relations between the critical exponents of the 8-vertex model can be derived by a mapping of this model onto the Luttinger model. The equivalence of this method to the one that connects the 8-vertex model to the Gaussian model is discussed. The Luttinger model is equivalent to the Gaussian model. Its operators are identified as vortex and spin-wave operators. The spin-wave operator cos4 is present in the critical 8-vertex Hamiltonian via an umklapp process. This explains the Kosterlitz-Thouless transition in the 6-vertex model, and resolves questions concerning the validity of the lattice continuum limit in the treatment by Luther and Peschel.