Renormalization group transformations for spin systems

We develop a systematic approach to perform real space renormalization group transformations of the "decimation type" using perturbation theory. This type of transformations beyond d = 1 is non-trivial even for the Gaussian model on the lattice. Such a transformation is constructed on a hypercubic lattice in arbitrary dimensions, and perturbation theory for spin models is developed around it. We check the formalism on the solvable O(N) symmetric Heisenberg chain. The decimations are especially useful to study models undergoing a continuous phase transition at zero temperature. Results for one class of such models, the D = 2 O(N) symmetric classical spins (N ≥ 3) for decimation with scale factor η = 2 (when one quarter of the points is left), are presented.