Perfect lattice action for asymptotically free theories

TL;DR

This paper discusses the development of a perfect lattice action for asymptotically free theories, which maintains continuum physics properties even on coarse lattices, improving accuracy and symmetry restoration.

Contribution

It introduces a method to construct short-ranged perfect lattice actions combining analytical and numerical techniques, demonstrated on the 2D non-linear sigma model.

Findings

Rotation symmetry is restored on coarse lattices

Spectrum becomes exact with the perfect action

Classical instanton solutions exhibit scale invariance

Abstract

There exist lattice actions which give cut--off independent physical predictions even on coarse grained lattices. Rotation symmetry is restored, the spectrum becomes exact and, in addition, the classical equations have scale invariant instanton solutions. This perfect action can be made short ranged. It can be determined by combining analytical calculations with numerical simulations on small lattices. We illustrate the method and the benefits on the $d=2$ non--linear $\sigma$--model.