# Iterating Block Spin Transformations of the O(3) Non-Linear Sigma-Model

## TL;DR

This paper investigates the iterative process of block spin transformations in the 2D O(3) non-linear sigma-model using Monte Carlo methods, aiming to understand the renormalization group flow and effective actions.

## Contribution

It introduces explicit computation of block spin effective actions with large parameter sets and compares different methods for determining effective couplings.

## Key findings

- Truncated effective actions show sizable scaling violations.
- Larger ansatz for effective action needed to reduce lattice artefacts.
- Renormalization group flow studied for various parametrizations.

## Abstract

We study the iteration of block spin transformations in the O(3) symmetric non-linear sigma-model on a two-dimensional square lattice with help of the Monte Carlo method. In contrast to the classical Monte Carlo Renormalization Group approach, we do attempt to explicitly compute the block spin effective actions. Using two different methods for the determination of effective couplings, we study the renormalization group flow for various parametrization and truncation schemes. The largest ansatz for the effective action contains thirteen coupling constants. Actions on the renormalized trajectory should describe theories with no lattice artefacts, even at small correlation length. However, tests with the step scaling function of Luescher et al. reveal that our truncated effective actions show sizable scaling violations indicating that the ansaetze are still too small.