Super-Instantons, Perfect Actions, Finite Size Scaling and the Continuum Limit

TL;DR

The paper explores the continuum limit of 2D O(N) lattice models, emphasizing the role of boundary conditions and fluctuations, and introduces a criterion to evaluate the proximity of lattice simulations to the continuum.

Contribution

It presents a new criterion for assessing how close lattice simulations are to the continuum limit, clarifies the role of boundary conditions, and challenges recent claims about super-instanton boundary conditions.

Findings

Fluctuations must be O(1) in the continuum limit, restoring symmetry.

The criterion helps identify lattice artefacts even in perfect actions.

Super-instanton boundary conditions do not require different renormalization, contrary to recent claims.

Abstract

We discuss some aspects of the continuum limit of some lattice models, in particular the $2D$ $O(N)$ models. The continuum limit is taken either in an infinite volume or in a box whose size is a fixed fraction of the infinite volume correlation length. We point out that in this limit the fluctuations of the lattice variables must be $O(1)$ and thus restore the symmetry which may have been broken by the boundary conditions (b.c.). This is true in particular for the so-called super-instanton b.c. introduced earlier by us. This observation leads to a criterion to assess how close a certain lattice simulation is to the continuum limit and can be applied to uncover the true lattice artefacts, present even in the so-called 'perfect actions'. It also shows that David's recent claim that super-instanton b.c. require a different renormalization must either be incorrect or an artefact of perturbation theory.