The schwinger model with perfect staggered fermions

Abstract

We construct a new perfect action for free staggered fermions, which is more local than the one obtained from the standard block average scheme. This pays off in superior properties after a short ranged truncation. This action is "gauged by hand" and tested in Schwinger model simulations by means of a new variant of hybrid MC. Using "fat links" for the gauge field, we obtain a tiny "pion" mass down to β < ∼ 1.5, and the "eta" mass follows very closely the prediction of asymptotic scaling.It has been observed that a good implementation of classically perfect actions provides an excellent level of improvement in the sense that lattice artifacts are drastically suppressed. However, the most successful applications have been limited to two dimensions so far, and they involve a huge number of couplings [1,2]. This implies that corresponding 4d actions are hardly applicable. Therefore it is crucial to achieve good improvement with rather few extra terms, and also this can be studied first in d = 2. However, the known classically perfect actions of Refs. [1,2] are not local enough for a very short ranged truncation to be sensible, hence -in view of d = 4 -different construction methods should be considered. Here we present a successful application of "gauging by hand", i.e. we use the truncated perfect couplings of the free fermion and insert gauge couplings by hand. This leads to a relatively modest overhead in the number of couplings compared to the standard action, but to an improvement on the same level as classical perfection. We conclude that this procedure is promising for d = 4. order to preserve the remnant chiral symmetry U (1) ⊗ U (1). This can be achieved by blocking each flavor separately [3]. As an improvement over the standard block average (BA) scheme [4], we propose "partial decimation" (PD) [5]. The free propagator ∆ BA is built by averaging source and sink in each block, whereas in the PD scheme one only averages the source and treats the sink by decimation, or vice versa. Both lead to local perfect actions for free staggered fermions, and in both cases we optimize the locality by tuning the RGT parameters. The optimization criterion is that the mapping down to d = 1, ∆ −1 (p 1 , 0, . . . , 0), only couples nearest neighbors. For the PD scheme this is the case for a simple δ function blocking at m = 0 [5]. However, that criterion can be fulfilled at arbitrary mass m. In d > 1 the PD scheme leads to a superior degree of locality, i.e. to a faster exponential decay of the couplings. Hence the truncation can be expected to be less harmful.The truncation itself should also be done with some care. We found mixed periodic boundary conditions to be optimal: we impose antiperiodicity over 6 lattice spacings in the direction of odd coupling distance, and periodicity over 6 lattice spacings in the other direction(s). The resulting couplings in d = 2 are given in the following Table; for d = 4 we refer again to Ref. [5]. There we also discuss massive fermions, including the case of non-degenerated flavors.

Renormalization group and perfect operators for stochastic differential equations

“…But there are other considerations as well [26,27]. One would like the operator to satisfy certain constraints that stipulate the correct behavior of the operator in the continuum limit.…”

“…The average constraint error is about 0.1% if one keeps up to ∼ 20 and ∼ 3 neighbors in the x and t directions respectively. An alternative approach [26,27] is to compute the perfect operator on a smaller lattice and then use this 'naturally' truncated perfect operator. In this way, the constraint is taken care of in the continuum limit.…”

The Schwinger model with perfect staggered fermions

“…The better quality of the truncated perfect action arising from the PD scheme is confirmed by the dispersion relation and by the thermodynamic scaling behavior [5,6]. In the latter case, the scaling region is extended by one order of magnitude compared to the standard action.…”

“…, 0), only couples nearest neighbors. For the PD scheme this is the case for a simple δ function blocking at m = 0 [5]. However, that criterion can be fulfilled at arbitrary mass m. In d > 1 the PD scheme leads to a superior degree of locality, i.e.…”

“…This can be achieved by blocking each flavor separately [3]. As an improvement over the standard block average (BA) scheme [4], we propose "partial decimation" (PD) [5]. The free propagator ∆ BA is built by averaging source and sink in each block, whereas in the PD scheme one only averages the source and treats the sink by decimation, or vice versa.…”