Decimation in more then one dimension

TL;DR

This paper develops a formalism for real space renormalization group transformations of the decimation type in multiple dimensions, applying it to free theories and asymptotically free models like 2D O(N) sigma models.

Contribution

It introduces a perturbative formalism for decimation-based renormalization in higher dimensions, extending beyond one-dimensional cases and applying it to specific models.

Findings

Formalism verified on solvable O(N) Heisenberg chain

Applied to 2D O(N) sigma models with decimation scale factor 2

Provides insights into asymptotically free theories

Abstract

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 $\sigma$ models ($N\ge 3$) for decimation with scale factor $\eta=2$ (when quarter of the points is left) are given as an example.