Locality with staggered fermions

TL;DR

This work analyzes the locality of defining a two-taste theory by taking the square root of the staggered fermion determinant. It shows analytically that the candidate local operator D = \sqrt{(M^{\dagger}M)_{e}} is non-local in the continuum, with a localization length r_{loc} that scales as ~1/m, and confirms this non-local behavior in the free theory via an explicit kernel, while numerical quenched lattice studies indicate a finite non-localization range set by the lightest hadron scale. The results imply that the square-root trick, as implemented through a naive local D, cannot yield a consistent local quantum field theory for two tastes, though the open question of a different local operator D remains. Consequently, current simulations relying on the square-root determinant require careful interpretation regarding locality and unitarity, despite correctly reproducing the Boltzmann weight.

Abstract

We address the locality problem arising in simulations, which take the square root of the staggered fermion determinant as a Boltzmann weight to reduce the number of dynamical quark tastes. A definition of such a theory necessitates an underlying local fermion operator with the same determinant and the corresponding Green's functions to establish causality and unitarity. We illustrate this point by studying analytically and numerically the square root of the staggered fermion operator. Although it has the correct weight, this operator is non-local in the continuum limit. Our work serves as a warning that fundamental properties of field theories might be violated when employing blindly the square root trick. The question, whether a local operator reproducing the square root of the staggered fermion determinant exists, is left open.

Figures (7)

• Figure 1: Decay of $\langle f(r) \rangle$ defined in eq. (\ref{['fr']}). The curves are normalized to be 1 at the distance $r_{\rm ref}/r_0=3.72$ corresponding to the location of the rightmost vertical dotted line.
• Figure 2: Effective localization ranges $r_{\rm loc}(r)$ at $\beta=6.0,\;am=0.01$ for different volumes. The vertical dotted lines correspond to distances from the source smaller than the minimal distances $\;r_{\rm min}(L)\approx L\;$ at which finite volume effects for $L/a=16,\;24$ become sizeable.
• Figure 3: The continuum limit of the upper bound on the localization range in the volume $Lm_{\rm G}\approx6$.
• Figure 4: The continuum limit of the localization range computed at two constant physical distances from the source.
• Figure 5: Comparison of the lattice eq. (\ref{['eq-rootx']}) and continuum eq. (\ref{['eq-rootcont']}) results.