Renormalization Group transformations of the decimation type in more than one dimension

Abstract

We develop a formalism for performing real space renormalization group transformations of the "decimation type" using low temperature perturbation theory. This type of transformations beyond $d=1$ is highly nontrivial even for free theories. We construct such a solution in arbitrary dimensions and develop a weak coupling perturbation theory for it. The method utilizes Schur formula to convert summation over decorated lattice into summation over either original lattice or sublattice. We check the formalism on solvable case of $O(N)$ symmetric Heisenberg chain. The transformation is particularly useful to study models undergoing phase transition at zero temperature (various $d=1$ and $d=2$ spin models, $d=2$ fermionic models, $d=3,4$ nonabelian gauge models...) for which the weak coupling perturbation theory is a good approximation for sufficiently small lattice spacing. Results for one class of such spin systems, the d=2 O(N) symmetric spin models ($N\ge 3$) for decimation with scale factor $η=2$ (when quarter of the points is left) are given as an example