The puzzle of apparent linear lattice artifacts in the 2d non-linear σ-model and Symanzik's solution

Abstract

Lattice artifacts in the 2d O(n) non-linear σ-model are expected to be of the form O(a 2 ), and hence it was (when first observed) disturbing that some quantities in the O(3) model with various actions show parametrically stronger cutoff dependence, apparently O(a), up to very large correlation lengths. In a previous letter [1] we described the solution to this puzzle. Based on the conventional framework of Symanzik's effective action, we showed that there are logarithmic corrections to the O(a 2 ) artifacts which are especially large (ln 3 a) for n = 3 and that such artifacts are consistent with the data. In this paper we supply the technical details of this computation. Results of Monte Carlo simulations using various lattice actions for O(3) and O(4) are also presented.

Topological lattice actions

“…This would contradict Symanzik's effective theory. However, as investigated in detail [59,60], the standard and the modified actions indeed have lattice artifacts of O(a 2 log 3 (a/L)), with large logarithmic corrections mimicking O(a) effects. In particular, Symanzik's theory correctly describes the observed lattice artifacts.…”

“…The step scaling function is affected by lattice artifacts which -for some time -seemed not to be described by Symanzik's effective theory. Interestingly, a recent careful study of the lattice artifacts has shown that large logarithms arise, and that Symanzik's theory does describe the lattice artifacts correctly [59,60]. Hence, one may conclude that the continuum limit of the finite-volume mass gap m(L) is finally well understood.…”

“…By measuring the mass gap m(2L) for these values, we have then determined the lattice value Σ(2, u 0 , a/L) = 2Lm(2L) of the step scaling function, which is known to approach the continuum limit σ(2, u 0 = 1.0595) = 1.26121035(2) [59,60]. Figure 5 compares the cut-off effects of Σ(2, u 0 , a/L) for the topological action with the standard and with two modified actions.…”

“…We attribute these deviations to corrections to the assumed O(a 2 ) behavior, similar to the ones discussed before for the mass gap. In order to fully understand the cut-off effects, an analytic analysis along the lines of [59,60], combined with numerical data closer to the continuum limit would be most welcome. Keeping this in mind, we still conclude that our current data confirm again that the topological action leads into the standard O(3) universality class.…”

An improved single-plaquette gauge action

“…Interestingly, in contrast to what Symanzik improvement suggests, the cut-off effects in the 2-d O(3) model appeared to be O(a) instead of O(a 2 ) [17,18]. A careful analysis of this apparent contradiction verified the Symanzik O(a 2 ) prediction, but showed that easily accessible lattice spacings are affected by large logarithmic corrections, which mimic O(a) behavior [19,20]. Recently, different lattice actions have been studied with the goal to improve the cut-off effects [21].…”