The Hierarchical Model and the Renormalization Group

The models tha t we consider are 1-dimensionM ferromagnets with a ra ther peculiar, nontranslat ionMly invar iant interact ion [1-4]. I t will be seen tha t the special form of this interact ion leads to a ra ther simple formulat ion of the renormalizat ion group equation. Apar t f rom the lack of t ranslat ion invariance these models are supposed to behave in m a n y respects like ordinary 1-dimensionM ferromagnets with (in the interesting ease) a long-range interact ion oi s t rength r -~, 1 < ~ < 2. Let n = 1, 2, ... label the points of a semi-infinite one-dimensionM lattice. On each site sits a spin ~, which is a real number a , e (-c~, c~). For each integer 10 we break the latt ice up in groups of 2~ consecutive sites; the m-th 10-group will consist of the sites