The Renormalization Group and its Finite Lattice Approximations

TL;DR

This paper explores finite lattice approximations to the Wilson Renormalization Group for spin models, focusing on methods to improve accuracy and applying them to compute critical properties in a two-dimensional sigma model.

Contribution

It introduces techniques to identify optimal renormalization group transformations for accurate critical quantity computations in lattice models.

Findings

Identified optimal RGTs for accurate critical exponent calculations

Demonstrated effective application to a 2D linear sigma model

Discussed advantages and limitations of finite lattice RGT approximations

Abstract

We investigate finite lattice approximations to the Wilson Renormalization Group in models of unconstrained spins. We discuss first the properties of the Renormalization Group Transformation (RGT) that control the accuracy of this type of approximations and explain different methods and techniques to practically identify them. We also discuss how to determine the anomalous dimension of the field. We apply our considerations to a linear sigma model in two dimensions in the domain of attraction of the Ising Fixed Point using a Bell-Wilson RGT. We are able to identify optimal RGTs which allow accurate computations of quantities such as critical exponents, fixed point couplings and eigenvectors with modest statistics. We finally discuss the advantages and limitations of this type of approach.