A classical large N hierarchical vector model in three dimensions: A nonzero fixed point and canonical decay of correlation functions

Abstract

We consider a hierarchical N-component classical vector model on a three-dimensional lattice Z3, for large N. The model differs from the usual one in that the kernel of the inverse Laplace operator is nontranslational invariant but has matrix elements which are positive and exhibit the same falloff as the inverse Laplacian in Z3. We introduce a renormalization group transformation and for N=∞, corresponding to the leading order of the 1/N expansion, we construct explicitly a nonzero fixed point for this transformation and also obtain some correlation functions. The two-point function has canonical decay. For 1≪N<∞, we obtain the fixed point and the two-point function in the first 1/N approximation. Canonical decay is still verified, in contrast to what is reported for the full model.