Scaling tests with dynamical overlap and rooted staggered fermions

TL;DR

This work performs a comprehensive scaling study comparing dynamical overlap and rooted staggered fermions in the 1-flavor Schwinger model. By analyzing a suite of observables—scalar condensate, topological susceptibility, partition function with Leutwyler-Smilga sum rules, heavy-quark potential, and determinant ratios—the authors assess universality and cutoff effects, introducing UV-filtering to improve taste-like artifacts. They find that, for finite sea-quark mass, both discretizations yield consistent continuum limits for most observables, though the chiral limit of the condensate exhibits non-commutativity for rooted staggered, and determinant ratios show $O(a^2)$ differences. The results reinforce that overlap fermions generally offer smaller discretization errors and clearer chiral behavior, while rooted staggered fermions can reproduce continuum physics at finite mass but raise questions about limit ordering and locality in 4D QCD.

Abstract

We present a scaling analysis in the 1-flavor Schwinger model with the full overlap and the rooted staggered determinant. In the latter case the chiral and continuum limit of the scalar condensate do not commute, while for overlap fermions they do. For the topological susceptibility a universal continuum limit is suggested, as is for the partition function and the Leutwyler-Smilga sum rule. In the heavy-quark force no difference is visible even at finite coupling. Finally, a direct comparison between the complete overlap and the rooted staggered determinant yields evidence that their ratio is constant up to $O(a^2)$ effects.

Figures (18)

• Figure 1: Low-energy spectrum of the unfiltered and two filtered versions of the Dirac operators $D^\mathrm{st}$ and $D^\mathrm{ov}$ [after chiral rotation as in (\ref{['condovereig2']})] on three typical configurations at $\beta\!=\!7.2$ and on a selected one where the overlap charge depends on the smearing level (rightmost panel).
• Figure 2: Bare overlap and rooted staggered ${N_{\!f}}\!=\!1$ condensate $\chi_\mathrm{sca}/e$ at $\beta\!=\!7.2$, plotted versus the quark mass. Here and in subsequent figures, three lines indicate a $1\sigma$-error band. The overlap result changes little, if a UV filtered Wilson operator is used instead of an unfiltered kernel. By contrast, in the staggered case this makes a big difference -- only the filtered variety shows a clear separation into a regime (here: $m/e\!>\!0.02$) where the staggered answer is meaningful, and a regime ($m/e\!<\!0.02$) where lattice artefacts overwhelm.
• Figure 3: Bare overlap condensate $\chi_\mathrm{sca}^\mathrm{ov}/e$ with $m\!=\!0$ at zero temperature (left) or finite temperature (right) versus $(ae)^2$. The three filtering levels have a universal continuum limit, and the extrapolation (\ref{['ansatz_zeromass']}) works well. The zero temperature result is consistent with (\ref{['schwinger']}).
• Figure 5: Left: Free condensates versus $1/\beta$ (in our zero temperature geometries, extended towards larger $\beta$), at fixed $m/e\!=\!0.1$. The fit to eqn. (\ref{['fit_free']}) uses the data below 0.4, i.e. $L\!\geq\!16$. Right: Difference of the free staggered and overlap condensate versus $1/\beta$. The data to the left of 0.4 are fitted with a $O(1/\beta)\!+\!O(1/\beta^{3/2})$ ansatz, shown together with its leading part.
• Figure 6: The "naive" subtracted condensates (\ref{['over_subt_nai']}, \ref{['stag_subt_nai']}) at $\beta\!=\!7.2$, together with $\chi_\mathrm{sca}$ and $\chi_\mathrm{free}$ for overlap (left) and staggered (right) fermions. Note the power-like IR-divergence in $\chi_\mathrm{subt}^\mathrm{ov/st}$.