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Theoretical issues with staggered fermion simulations

Stephan Durr

TL;DR

The paper analyzes whether dynamical staggered fermions with rooting, $\det^{1/4}(D_{\mathrm{st}})$, reproduce the correct nonperturbative continuum limit and emphasizes locality as the key nonperturbative hurdle. It surveys the staggered action’s taste structure, presents several free-theory constructions for a local candidate action $D_{\mathrm{ca}}$ with the correct determinant, and surveys spectral, topological, and effective-field-theory evidence from 4D and 2D theories. Results show encouraging but inconclusive support for rooting: improved actions and filtering reduce taste breaking, overlap-staggered determinant correlations improve with finer lattices, and SXPT can describe rooted data, yet no general proof exists. The work underscores the need for a proven local, doubler-free candidate action or a robust counterexample and highlights the practical implications for lattice QCD with ${N_f}=2+1$ quarks.

Abstract

The legality of the "rooting trick" in dynamical staggered fermion simulations is discussed, i.e. whether the theory with the Boltzmann weight $\det^{1/4}(D_\mathrm{st})$ yields the right continuum limit. Since the problem is unsolved, pieces of evidence in favor and against are collected and examined.

Theoretical issues with staggered fermion simulations

TL;DR

The paper analyzes whether dynamical staggered fermions with rooting, , reproduce the correct nonperturbative continuum limit and emphasizes locality as the key nonperturbative hurdle. It surveys the staggered action’s taste structure, presents several free-theory constructions for a local candidate action with the correct determinant, and surveys spectral, topological, and effective-field-theory evidence from 4D and 2D theories. Results show encouraging but inconclusive support for rooting: improved actions and filtering reduce taste breaking, overlap-staggered determinant correlations improve with finer lattices, and SXPT can describe rooted data, yet no general proof exists. The work underscores the need for a proven local, doubler-free candidate action or a robust counterexample and highlights the practical implications for lattice QCD with quarks.

Abstract

The legality of the "rooting trick" in dynamical staggered fermion simulations is discussed, i.e. whether the theory with the Boltzmann weight yields the right continuum limit. Since the problem is unsolved, pieces of evidence in favor and against are collected and examined.

Paper Structure

This paper contains 9 sections, 26 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Review: Staggered action and taste representation
  3. Problem: rooting versus locality
  4. Free case: four constructions
  5. Interacting theory: $\mathrm{spec}(D_\mathrm{st})$ in 4D
  6. Interacting theory: $\chi_\mathrm{sca}\,,\chi_\mathrm{top}$ in 2D
  7. Correlation of $\det^{1/{N_{t}}}(D_{\mathrm{st},m})$ and $\det(D_{\mathrm{ov},m})$
  8. Staggered Chiral Perturbation Theory
  9. Summary

Figures (13)

  • Figure 1: Staggered and overlap eigenvalues on 4 selected configurations in 2D. On benevolent configurations the improved taste symmetry achieved through filtering results in a 2-fold (4-fold in 4D) near-degeneracy and the right number of near-zero modes. To aid comparison the low-energy overlap spectrum has been mapped onto the imaginary axis with a stereographic projection. Figure from Durr:2003xs.
  • Figure 2: Localization function $f(r)$ in physical units versus $r/r_0$ for the "candidate" action of Bunk:2004br. The local logarithmic derivative (the effective $r_\mathrm{loc}$) scales and the operator is thus non-local. Figure from Bunk:2004br.
  • Figure 3: Eigenvalue spectra of two "candidate" operators $D_\mathrm{ca}$ on the blocked lattice in 2D which reproduce $\det^{1/2}(D_\mathrm{st})$ of the massless staggered action in the free theory. On the left $D_\mathrm{ca}$ is exclusively optimized for locality, on the right the (standard) Ginsparg Wilson relation is imposed as a constraint. Figure from Maresca:2004me.
  • Figure 4: Chirality versus eigenvalue of staggered eigenmodes. On coarse lattices there is no distinctive signature, but on somewhat finer lattices rather aggressive filtering reveals continuum like features. Near-zero modes are predominantly chiral, non-zero modes are not. Figure from Talk_Hasenfratz.
  • Figure 5: Chirality and eigenvalue versus "serial number" of the lowest few staggered eigenmodes of the positive staggered half-spectrum. The left panel is for Wilson glue, the right one for the Symanzik action, with $a\!\simeq\!0.1\,\mathrm{fm}$ in either case. The staggered action with "HISQ" filtering sees $|q|\!=\!2$. Figure from Follana:2004sz.
  • ...and 8 more figures