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

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.

14 pages, latex, no figures

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• Perfect lattice action for asymptotically free theories
• Lattice QCD on Small Computers
• Asymptotic Scaling in the Two-Dimensional $O(3)$ $\sigma$-Model at Correlation Length $10^5$
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• Existence of Algebraic Decay in Nonabelian Ferromagnets

Cited by in corpus (3)

• Continuum Limit of $2D$ Spin Models with Continuous Symmetry and Conformal Quantum Field Theory
• Lattice artefacts and the running of the coupling constant
• Testing Asymptotic Scaling and Nonabelian Symmetry Enhancement