Decimation in more then one dimension

We develop a formalism for performing real space renormalization group transformations of the "decimation type" using perturbation theory. The type of transformations beyond $d=1$ is nontrivial even for free theories. We check the formalism on solvable case of $O(N)$ symmetric Heisenberg chain. The transformation is particularly useful to study asymptotically free theories. Results for one class of such models, the d=2 O(N) symmetric $σ$ models ($N\ge 3$) for decimation with scale factor $η=2$ (when quarter of the points is left) are given as an example.

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