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

TL;DR

This study uses high-statistics Monte Carlo simulations to examine the phase structure of two-dimensional O(n) non-linear sigma models, providing evidence supporting asymptotic freedom in the O(8) case.

Contribution

It offers new high-precision Monte Carlo data on the mass gap and susceptibility, clarifying the phase behavior of O(n) models and supporting the asymptotic freedom hypothesis.

Findings

Results support asymptotic freedom in O(8) model

No evidence of Kosterlitz-Thouless transition observed

Provides high-precision data for mass gap and susceptibility

Abstract

We have performed a high statistics Monte Carlo simulation to investigate whether the two-dimensional O(n) non-linear sigma models are asymptotically free or they show a Kosterlitz- Thouless-like phase transition. We have calculated the mass gap and the magnetic susceptibility in the O(8) model with standard action and the O(3) model with Symanzik action. Our results for O(8) support the asymptotic freedom scenario.

Equations

$$S=\frac{\beta }{2}\int {\mathrm{d}}^{2}x{\left({\partial }_{\mu }\vec{\varphi }\right)}^2$$

$$\xi =C_{\xi }{\left(\frac{1}{2\pi \beta \Delta }\right)}^{\Delta }{e}^{2\pi \beta \Delta }\left(1+\sum _{k=1}^{\infty }\frac{a_k}{{\beta }^k}\right)$$

$$C_{\xi }={\left(\frac{e}{8}\right)}^{\Delta }\Gamma (1+\Delta )2^{-5/2}\exp \left(-\frac{\pi \Delta }{2}\right).$$

$$\chi =C_{\chi }{\left(\frac{1}{2\pi \beta \Delta }\right)}^{\Delta (n+1)}{e}^{4\pi \beta \Delta }\left(1+\sum _{k=1}^{\infty }\frac{b_k}{{\beta }^k}\right)$$

$$R_{PT}\equiv \frac{\chi }{{\xi }^2}(2\pi \beta \Delta {)}^{\frac{n-1}{n-2}}\left(1+\sum _{k=1}^{\infty }\frac{d_k}{{\beta }^k}\right)$$

$$R_{KT}\equiv \frac{\chi }{{\xi }^{2-\eta }}$$

Full text

1 INTRODUCTION

The 2-dimensional $O(n)$ non-linear $\sigma$-model is defined by the action

[TABLE]

together with the condition ${\vec{\phi}}(x)^{2}=1$ for all spacetime points $x$. In this equation $\beta$ is the inverse of the bare coupling constant.

Perturbation theory $(PT)$ predicts that this model is asymptotically free for $n\geq 3$. In particular the exponential correlation length $\xi$ on the lattice must scale as

[TABLE]

where $a_{k}$ are the corrections to universal scaling. Here $\Delta\equiv 1/(n-2)$. $C_{\xi}$ is a non-perturbative constant which for the standard action equals [1]

[TABLE]

We define the magnetic susceptibility $\chi$ as the two-point correlation function at zero momentum. It scales as

[TABLE]

where again $C_{\chi}$ is a non-perturbative constant. From equations (2) and (4) we conclude that in $PT$ the ratio

[TABLE]

tends to $C_{\chi}/C_{\xi}^{2}$ as we approach the continuum limit, $\beta\rightarrow\infty$. The corrections to asymptotic scaling $d_{k}$ depend on $\{a_{k}\}$ and $\{b_{k}\}$.

In a series of papers [2] another scenario has been put forward for the model defined in (1). Under reasonable hypothesis the authors prove that there is no mass gap and that this model must undergo a Kosterlitz-Thouless-like ($KT$) phase transition at finite beta, $\beta_{KT}$. This implies that the ratio

[TABLE]

should be constant as one approaches $\beta_{KT}$ from below. Here $\eta$ is a critical exponent. For the $O(2)$ model this exponent is $\eta=1/4$. In [3] the authors show that the $O(3)$ model with the standard action on the lattice and $\eta=1/4$ gives a constant for $R_{KT}$ while the data for $R_{PT}$ displays a clear drop.

Here we will show a progress report from an extensive simulation performed on the $O(8)$ model with standard action and the $O(3)$ model with the tree-level improved Symanzik action [4]. If the constancy of $R_{KT}$ for the $O(3)$ model is a genuine physical effect, then also for the Symanzik action we should see such a behaviour. The full account of our results with better statistics and using more corrections to asymptotic scaling can be found in [5].

2 SIMULATIONS

In our simulations we have used the Wolff algorithm [6] for the updatings as well as improved estimators to measure the correlation length and magnetic susceptibility. We have performed several millions of measurements for both quantities and verified the absence of autocorrelations.

To calculate the correlation length we have measured the second moment $\xi^{(2)}$. The ratio $\xi^{(2)}/\xi$ is less than few parts per mille, so within our statistical errors, we can use the formulae (2-3).

We have chosen large enough lattice sizes $L$ to keep finite-size effects under control. The ratio $L/\xi$ is $7-10$. We have checked that these effects are few parts per mille.

The largest systematic error comes from the deviation from universal scaling of our data. These corrections are known up to 4 loops for the standard action and up to 3 loops for the Symanzik action [7].

3 RESULTS

3.1 The $O(3)$ model

In figure 1 we show the results for $R_{KT}$ in the $O(3)$ model with Symanzik action. They have better statistics than those of reference [3]. In constrast with [3] our data are not constant.

We do not show here (see [5]) the data for $R_{PT}$. Again it is not constant although it displays better scaling than for the standard action [3]. The fits for $C_{\xi}$ and $C_{\chi}$ agree with the prediction (3) and large-$n$ calculations within $15-20\%$.

Assuming finite-size scaling, it has been shown that this model presents asymptotic scaling starting from $\xi\approx 10^{5}$ [8]

3.2 The $O(8)$ model

In figure 2 we show the ratio between the non-perturbative constant $C_{\xi}^{\rm MC}$ as computed from our Monte Carlo data and the prediction (3) which for the $O(8)$ model is $C_{\xi}=0.10544$. If $PT$ is correct and asymptotic scaling holds, this ratio should be equal to 1 (up to $\sim$0.1 per mille because we measured the second moment $\xi^{(2)}$).

We show the data as obtained from the 2, 3 and 4-loop approximation in eq. (2). The data in the scheme of the energy [9] at 2 and 3-loop are also shown (we have used the energy measurements of reference [10]). All data seem to converge towards the $PT$ prediction. A careful analysis of the next coefficients in the $1/n$ expansion suggests that further corrections should have small effects. The 4-loop data in this figure agree with (3) within 0.5%.

In figure 3 the data for $R_{KT}$ are shown. The data are far from constant. We show the results for two values of $\eta$: $\eta=0.25$ is the upper set of data (black circles) and $\eta=0.22$ is the lower set (white circles).

The solid and dashed lines are the $PT$ predictions for $R_{KT}$ assuming $C_{\chi}=0.103$ (this is the value obtained from a best fit performed on our data for $\chi$; it agrees with large-$n$ estimates within 1%) and the prediction (3) for $C_{\xi}$. We see that not only the curves are not constant but also that $PT$ explains well its non-constancy.

In figure 4 we show the data for $R_{PT}$. The upper set of data is the lowest order prediction. Further corrections stabilize the result. The result clearly converges to a constant. Physical scaling is well reached at the largest correlation lengths. The corrections to universal scaling converge surprisingly well.

The solid horizontal line is the $PT$ prediction for the constant by using $C_{\chi}=0.103$ and the prediction (3) for the correlation length. We conclude that our data are in fair agreement with $PT$.

In conclusion our data do not support either $KT$ or $PT$ for the $O(3)$ model but they show clear agreement with $PT$ for the $O(8)$ model. In the $KT$ scenario one should explain why $PT$ works so well for large $n$ and large ratios $L/\xi$.

This work has benefited from many stimulating conversations with Andrea Pelissetto.