A Study of Finite Temperature Gauge Theory in (2+1) Dimensions

TL;DR

This paper investigates the finite temperature phase transition in (2+1)-dimensional SU(2) and SU(3) gauge theories, determining critical parameters and string tension to understand their thermodynamic behavior.

Contribution

It provides new measurements of critical couplings, exponents, and string tension in (2+1)D gauge theories, enhancing understanding of their finite temperature properties.

Findings

Critical couplings and exponents determined for SU(2) and SU(3).

Wilson loops measured at zero temperature with improved signal.

Calculated ratio T_c/√σ to characterize the phase transition.

Abstract

We determine the critical couplings and the critical exponents of the finite temperature transition in SU(2) and SU(3) pure gauge theory in (2+1) dimensions. We also measure Wilson loops at $T=0$ on a wide range of $\beta$ values using APE smearing to improve the signal. We extract the string tension $\sigma$ from a fit to large distances, including a string fluctuation term. With these two entities we calculate $T_c/\sqrt{\sigma}$.

Tables

Table 1. Table 1: The critical couplings and critical exponents for SU(2) and SU(3) in (2+1) dimensions.

SU(2)	${\beta }_{c,\infty }$	$\beta /\nu$	$\gamma /\nu$	$1/\nu$
$N_{\tau }=4$	6.52 (3)	0.14 (3)	1.70 (4)	0.9 (-1)(+3)
$N_{\tau }=6$	9.55 (4)	0.17 (3)	1.68 (7)	1.3 (4)
RM		0.125	1.75	1.0
SU(3)	${\beta }_{c,\infty }$	$\beta /\nu$	$\gamma /\nu$	$1/\nu$
$N_{\tau }=2$	8.155 (15)	0.12 (10)	1.71 (3)	1.11 (25)
$N_{\tau }=4$	14.74 (5)	0.12 (5)	1.72 (7)	1.14 (24)
$N_{\tau }=6$	21.34 (4)(11)	0.14 (2)	1.69 (5)	1.11
RM		0.1333	1.733	1.200

Equations

$$g_{3B}^2=g_3^2+a{g}_3^4+\cdots ,\beta =\frac{2N_c}{a{g}_{3B}^2}$$

$$L=\frac{1}{N_{\sigma }^2}\sum _x\frac{1}{N_c}\text{mbox}Tr\prod _{\tau =1}^{N_{\tau }}{U}_{\nu =0}(\tau ,x).$$

$$\begin{gathered} {\chi }_L=N_{\sigma }^2(⟨L^2⟩-⟨|L|{⟩}^2) \\ {g}_r=\frac{⟨L^4⟩}{⟨L^2{⟩}^2}-3. \end{gathered}$$

$${\chi }_L=N_{\sigma }^{\frac{\gamma }{\nu }}(c_0+(c_1+c_2{N}_{\sigma }^{-y_i})x{N}_{\sigma }^{1/\nu }+c_3{N}_{\sigma }^{-y_i}),$$

$$\ln {\chi }_L=\ln (c_0)+\gamma /\nu \ln (N_{\sigma }).$$

$$V_S(R)=\log \frac{⟨\mathrm{W}(R,S)⟩}{⟨\mathrm{W}(R,S+1)⟩},$$

$$V(R)=\underset{T\to \infty }{\lim }{V}_S(R).$$

$$\mathrm{a}\sqrt{\sigma }=\frac{c_0}{\beta }+\frac{c_1}{{\beta }^2}+\frac{c_2}{{\beta }^3}+\cdots$$

$$T_{c}/\sqrt{\sigma}=\sqrt{\frac{3}{\pi(d-2)}}=\cases{0.977&d = 3\cr 0.691&d = 4}$$

Full text

1 Introduction

The determination of the deconfinement temperature $T_{c}$ is an important task in SU(N) gauge theories. The Nambu-Goto string model [1] predicts a value for $T_{c}$ in units of the zero temperature string tension $\sqrt{\sigma}$ depending only on the dimension but not on the group. This has been tested for d=4 and good agreement has been found for N=3 and especially for N=2 [2]. Now we verify the prediction for $T_{c}/\sqrt{\sigma}$ in 3 dimensions again for the SU(2) and SU(3) case. We determine the critical exponents to check the universality class of the model. These exponents can be compared with those from the reduced model (RM), which is in the same universality class as the 2d Ising (SU(2)) and the 2d three-state-Potts (SU(3)) model.

We measured $\beta_{c}$ and the critical exponents on various asymmetric $N_{\sigma}^{2}\times N_{\tau}$ lattices and $\sqrt{\sigma}$ on a symmetric $32^{3}$ lattice using the standard Wilson action.

In (2+1) dimensions $g_{3}^{2}$ has a dimension and sets the scale. Then the bare lattice coupling is given by

[TABLE]

2 The Phase Transition

The order parameter for deconfinement on infinite volume lattices is the Polyakov loop,

[TABLE]

To find the critical coupling, we look at the Polyakov loop $\langle|L|\rangle$, the susceptibility $\chi_{L}$ and the Binder cumulant $g_{r}$

[TABLE]

We use the position of the maximum of the susceptibility to fix a pseudo critical point on a given lattice. We ran simulations on a wide range of spatial lattice sizes ranging from $N_{\sigma}=16$ to $96$ for $N_{\tau}=2$, $N_{\sigma}=16$ to $128$ for $N_{\tau}=4$ and $N_{\sigma}=48$ to $192$ for $N_{\tau}=6$ at couplings close to the expected infinite spatial volume critical coupling.

The susceptibility is shown in figure 1 for SU(3) and $N_{\tau}=4$. The interpolated lines are calculated by the multi histogram density of states method (DSM).

To find the critical coupling on an infinite spatial lattice one can extrapolate with Finite Size Scaling (FSS) ansatz from the values found in the susceptibility and the Binder cumulant, or use the $\chi^{2}$ Method [3].

The latter uses the FSS formula e.g. for the susceptibility111For computational reasons we better take the observable $\chi_{L}=N_{\sigma}^{2}\langle L^{2}\rangle$. See ref. [4] expanded around the critical coupling

[TABLE]

where $x=\frac{\beta_{c,\infty}-\beta}{\beta}$ is the reduced temperature that vanishes at the critical coupling, $c_{i}$ are coefficients and $y_{i}$ irrelevant exponents. Neglecting corrections $N_{\sigma}^{-y_{i}}$ and taking the logarithm we find exactly at $\beta_{c,\infty}$

[TABLE]

Elsewhere we expect deviations from this as can be seen from eq. 4. Thus in a plot of $\ln\chi_{L}$ vs. $\ln N_{\sigma}$ for fixed $\beta$ the points should lay on a straight line if (and only if) $\beta=\beta_{c,\infty}$.

Performing a linear fit for every $\beta$ value the one with the lowest $\chi^{2}/dof$ indicates $\beta_{c,\infty}$.

In figure 2 the result is shown for the susceptibility for SU(3), $N_{\tau}=4$. Plotted is $\gamma/\nu$ vs. $\beta$ (left scale) and $\chi^{2}$ vs. $\beta$ (right scale). The critical coupling is determined as the $\beta$ value at the minimum of the $\chi^{2}/dof$ curve. An estimate for the error in $\beta_{c,\infty}$ is the min$(\chi^{2})+1$, this also gives the error on $\beta/\nu$.

In table 1 the results for the critical couplings and exponents are summarized.

3 The String Tension

We measured Wilson loops on a symmetric $32^{3}$ lattice. In SU(2) we simulated only at the critical couplings for $N_{\tau}=4,6$. In SU(3) we analyzed the potential within a wide range of $\beta$ values ranging from 8.15 to 50. We applied APE smearing [5] to maximize the signal/noise ratio [6].

With the standard estimator

[TABLE]

we get the potential $V(R)$ in the limit of large $S$

[TABLE]

We used the function $V(R)=V_{0}+\frac{\alpha}{R}+\sigma R$ for three and two parameter fits. In the latter case $\alpha$ was fixed to $\pi/24$ which is the value for string fluctuations. This ansatz gave the best and most stable description of the potentials. We performed the fit from an $R_{min}$ to neglect short distance Coulomb behaviour.

From eq. 1 we can assume the following scaling behaviour

[TABLE]

In figure 3 we plot $\beta{\rm a}\sqrt{\sigma}$ versus $1/\beta$. The linear fit only contains the six lower data points. To describe $\beta{\rm a}\sqrt{\sigma}$ in the whole range of $\beta$ values one obviously needs a quadratic fit.

4 $T_{c}/\sqrt{\sigma}$

In the case of SU(3) we interpolate $\sqrt{\sigma}$ for the values $\beta_{c,\infty}$ for the various $N_{\tau}$ from equation 8. For SU(2) we calculate $T_{c}/\sqrt{\sigma}$ direct from the string tension obtained at the critical couplings.

Within the Nambu-Goto string model one finds

[TABLE]

The results obtained for $T_{c}/\sqrt{\sigma}$ in (2+1) dimensions for SU(2) and SU(3) from our lattice calculations are given below

[TABLE]

In figure 4 we plot $T_{c}/\sqrt{\sigma}$ vs. $\sigma a^{2}$ for SU(2) and SU(3) in 3 and 4 dimensions. The lines give an idea of the continuum extrapolation. The arrows mark the values predicted from the Nambu-Goto string model.

5 Conclusions

The critical couplings are well determined for SU(2) and SU(3) in 2+1 dimensions for $N_{\tau}=2,4,6$. They agree for SU(2), $N_{\tau}=4$ and SU(3), $N_{\tau}=2$ with the literature.

The critical exponents are in agreement with the values predicted from the reduced model. They are definitly not the meanfield exponents.

In SU(3) the string tension is well determined on a wide range of $\beta$ values and can be described by a cubic fit in $1/\beta$.

For SU(3) in 3 dimensions $T_{c}/\sqrt{\sigma}$ is in excellent agreement with the Nambu-Goto string model.

For SU(2) $T_{c}/\sqrt{\sigma}$ is 7% off the predicted value.

Acknowledgement We gratefully acknowledge support from the DFG under contract Pe 340/3-3. Part of the calculations were performed on the DFG-Quadrics at Bielefeld (Pe 340/6-2).