Reply to A. Patrascioiu's and E. Seiler's comment on our paper "The two-phase issue in the O(n) non-linear sigma-model: a Monte Carlo study"

TL;DR

This paper is a response to comments on a previous Monte Carlo study of the two-phase issue in the O(n) non-linear sigma-model, clarifying and defending the original findings.

Contribution

It provides a detailed reply addressing critiques of the Monte Carlo analysis in the context of the two-phase problem in the O(n) sigma-model.

Findings

Clarifies the original Monte Carlo results

Addresses specific critiques raised by Patrascioiu and Seiler

Reinforces the validity of the initial conclusions

Abstract

We reply to a comment by A. Patrascioiu and E. Seiler appeared in hep-lat/9608138 on our paper hep-lat/9608002.

Full text

IFUP-TH 56/96

September 1996

Reply to A. Patrascioiu’s and E. Seiler’s comment on our paper

The two-phase issue in the $O(n)$ non-linear ** $\sigma$-model: a Monte-Carlo study**

B. Allés, A. Buonanno and G. Cella

Dipartimento di Fisica and INFN,

Piazza Torricelli 2, 56126 Pisa, Italy

Our paper [1] has motivated a comment [2] by A. Patrascioiu and E. Seiler which we reply in this note. The remarks in [2] concern three statements that the authors select from our paper:

i) “The results for $O(8)$ support the asymptotic freedom scenario”.

The authors in [2] recall that at small $\tilde{\beta}\equiv\beta/N$ the $1/N$ expansion is Borel summable [3]. No rigorous proof has been given concerning the behaviour of the series at large $\tilde{\beta}$. In [2] it is not shown if our working $\beta$’s ($\beta\sim 4.6-6.5$) lie in the “small” or “large” $\tilde{\beta}$ region when $N=8$ but at least we can say that if the results from our Monte Carlo data fit so nicely the $PT$ predictions for the $O(8)$ model (within few per mille for the mass gap and susceptibility predictions, see [4]) there is reasonable room to think that the exact $O(8)$ model (what we simulate) has a critical point at $g=0$ and the set of $PT$ predictions are correct. If our data had to agree with some prediction other than the $PT$ set of predictions for the $O(8)$ model then the small difference between our result for the mass gap and the P. Hasenfratz et al. [5] prediction, 0.5% (compared for instance to the difference of almost 30% between the $n=8$ and the $n=\infty$ predictions) would become an intriguing challenge. Considering it as an accident is a matter of feelings.

ii) “Assuming finite-size scaling (FSS), it has been shown that $O(3)$ presents asymptotic scaling starting from $\xi=10^{5}$”.

This statement is true: Assuming FSS it has been shown that the $O(3)$ model presents asymptotic scaling within few per cent at large correlation lengths. We are aware of the validity of $PT$ whenever the limit $L/\xi\to 0$ holds, where $L$ is the lattice size and $\xi$ any correlation length, (see for instance [6]). For this reason we made our simulations at large values of the previous ratio, $L/\xi\sim 7-10$. Therefore the second comment of [2] does not apply to us.

iii) “The $O(3)$ model with Symanzik action does not show KT behavior”.

Strangely enough we have not written this sentence in our paper [1]. Maybe the authors had in mind some of the following sentences that do appear in the paper:

1. (1.)

“If the constancy of $R_{KT}$ … is a genuine physical effect, then also for the Symanzik action we should see such a behaviour”. 2. (2.)

“… our data [for the ratio $R_{KT}$] are not constant”. 3. (3.)

“… our data [for $O(3)$] do not support either $KT$ or $PT$”.

(1.) The tree-level improved Symanzik action (although invented in the $PT$ context) is as good as any other action for describing the $O(n)$ models on the lattice and it does not assume the validity of $PT$. Both the Lipschitz action (see for instance [7]) and the Symanzik action are in the same universality class as the standard action.

(2.) Figure 1 in [1] clearly shows that the $KT$ ratio is not constant, in contrast to what happens for the standard action [8]. We stress the fact that our data have better statistics. Clearly a similar high-statistics simulation for the standard action is worth doing.

(3.) We clearly say in [1] that the data for the mass, magnetic susceptibility and ratio $R_{PT}$ for the $O(3)$ model satisfy the $PT$ predictions only within $15\%$, [4]. This fact, together with point (2.), led us to conclude that our data do not support either $KT$ or $PT$. We cannot say more than this for the $O(3)$ model.