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Perfect lattice action for asymptotically free theories

P. Hasenfratz, F. Niedermayer

TL;DR

This work demonstrates that in asymptotically free theories there exist lattice actions—fixed-point actions—that are effectively free of lattice artefacts even on coarse grids. By combining analytical FP equations with targeted numerical minimization, Hasenfratz and Niedermayer construct a short-range FP action for the $d=2$ $O(3)$ sigma model and show it yields cut-off independent predictions and rotation-symmetric correlators. They further develop a practical parametrization using a small set of couplings, validate it on small lattices, and show that RG blocking can preserve locality at finite $\beta$, enabling the exploration of the renormalized trajectory with minimal artefacts. This framework offers a path toward artifact-free simulations in more complex theories, including gauge theories, via near-perfect actions and efficient finite-$\beta$ RG transformations.

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 $σ$--model.

Perfect lattice action for asymptotically free theories

TL;DR

This work demonstrates that in asymptotically free theories there exist lattice actions—fixed-point actions—that are effectively free of lattice artefacts even on coarse grids. By combining analytical FP equations with targeted numerical minimization, Hasenfratz and Niedermayer construct a short-range FP action for the sigma model and show it yields cut-off independent predictions and rotation-symmetric correlators. They further develop a practical parametrization using a small set of couplings, validate it on small lattices, and show that RG blocking can preserve locality at finite , enabling the exploration of the renormalized trajectory with minimal artefacts. This framework offers a path toward artifact-free simulations in more complex theories, including gauge theories, via near-perfect actions and efficient finite- RG transformations.

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 non--linear --model.

Paper Structure

This paper contains 18 sections, 49 equations, 1 figure, 6 tables.

Table of Contents

  1. Introduction and Summary
  2. The fixed point and the fixed point action
  3. The equation for ${\cal A}^*$
  4. ${\cal A}^*$ as a perfect classical theory on the lattice
  5. Parametrization
  6. The determination of $\rho(r)$
  7. The properties of $\rho(r)$; the perfect Laplace operator on the lattice
  8. The determination of $c(n_1,n_2,n_3,n_4)$
  9. The numerical procedure to solve the FP equation
  10. The properties of ${\cal A}^*$ on coarse configurations
  11. Fixing the couplings in this pilot study in O(3)
  12. Results obtained by simulating the FP action
  13. Cut--off dependence of the running coupling
  14. The two--point function
  15. RG transformation at finite $\beta$
  16. ...and 3 more sections

Figures (1)

  • Figure 2: Flow of the couplings under RG transformation in the O($N$) non--linear $\sigma$--model