Finite Temperature Properties of SO(3) Lattice Gauge Theories and their implications for the continuum limit

TL;DR

This paper investigates the finite temperature behavior of SO(3) lattice gauge theories, revealing metastable states and their relation to phase transitions, with implications for understanding the continuum limit and symmetry properties.

Contribution

It demonstrates the existence of metastable states in SO(3) lattice gauge theory and analyzes their connection to phase transitions and continuum limit implications.

Findings

Metastable states are related to bulk and finite temperature transitions.

Polyakov line in the adjoint representation traces metastable states.

Second order finite temperature transition is compatible with first order in SO(3).

Abstract

It is shown that $SO(3)$ lattice gauge theory on finite size lattices has metastable states related to the ground states of both the bulk transition and the finite temperature transition. The Polyakov line variable in the adjoint representation of $SU(2)$ is used to trace the origin of these metastable states. It is also argued that a second order finite temperature transition in the continuum theory is not inconsistent with the first order transition in $SO(3)$ lattice gauge theory and the absence of a $Z(2)$ global symmetry.

Equations

$$L_f=T{r}_{f}Pexpi{\int }_0^{\beta }d{x}_4{A}_0(\vec{x},x_4)$$

$$Z_2:L_f\to -L_f$$

$$\sum _p\left((\beta /2)T{r}_{f}U(p)+({\beta }_a/3)T{r}_{a}U(p)\right)$$

$$⟨L_f⟩=2\cos (\theta /2)$$

$$⟨L_a⟩=1+2\cos (\theta )$$

$$U=T{r}_{f}Pexpi{\int }_0^{\beta }d{x}_4{A}_0(\vec{x},x_4)$$

$$U\to V(\vec{x},0)U{V}^{†}(\vec{x},0)$$

Full text

**FINITE TEMPERATURE PROPERTIES OF SO(3) LATTICE GAUGE

THEORY AND THEIR IMPLICATIONS FOR THE CONTINUUM THEORY**

SRINATH CHELUVARAJA111e-mail:srinath@imsc.ernet.in

and

H.S. SHARATHCHANDRA222e-mail:sharat@imsc.ernet.in

The Institute of Mathematical Sciences

Madras - 600 113, INDIA

**ABSTRACT

**It is shown that $SO(3)$ lattice gauge theory on finite size lattices has metastable states related to the ground states of both the bulk transition and the finite temperature transition. The Polyakov line variable in the adjoint representation of $SU(2)$ is used to trace the origin of these metastable states. It is also argued that a second order finite temperature transition in the continuum theory is not inconsistent with the first order transition in $SO(3)$ lattice gauge theory and the absence of a $Z(2)$ global symmetry.

PACS numbers:12.38Gc,11.15Ha,05.70Fh,02.70g

There are many arguments that the Yang Mills (Y-M) theory undergoes a second order phase transition into a deconfined phase at finite temperature. The strong coupling limit of $SU(2)$ lattice gauge theory at $T\neq 0$ can be rewritten as a spin model with a global $Z(2)$ invariance [1]. At high temperatures, this symmetry is spontaneously broken. Monte Carlo simulations of $SU(2)$ lattice gauge theory (LGT) have confirmed that this result is not an artifact of the strong coupling limit and further indicate that the transition is of second order [2]. The order parameter for the transition is the Wilson-Polyakov line

[TABLE]

in the language of the continuum theory. Here the subscript $f$ indicates that the trace is taken in the fundamental representation. The $Z(2)$ group is related to the centre of the gauge group $SU(2)$ and its non trivial element has the action

[TABLE]

It has been argued that this transition should be in the universality class of the 3-d Ising model [3]. This is also supported by simulations.

Simulations of LGT’s with mixed actions have cast doubt on the above picture [4]. The Bhanot-Creutz [5] model has the action

[TABLE]

which is a sum of the actions in the fundamental and the adjoint representations. This LGT exhibits distinct phases even at $T=0$ (Fig.1). This bulk transition has been traced to a condensation of $Z_{2}$ monopoles in the small $\beta_{a}$ region [6]. There is a line of first order transitions with an endpoint of second order transition which stops short of intersecting the $\beta_{a}=0$ line. Study of this model [4] at $T\neq 0$ seems to suggest that the line of finite temperature transitions is continuously connected to the line of bulk first order transitions. This has led to a variety of contradictory interpretations in the literature. (i) It has been proposed that the bulk transition may not be really present and the effect is simply that of the finite temperature transition in simulations with finite lattices. (ii) The opposite scenario that the Y-M theory may not have a finite temperature transition at all has also been considered.(iii) The possibility that the Y-M theory has a first rather than a second order transition is also supposed.(iv) On the other hand, some studies, especially of related models with $SU(3)$ and $SU(4)$ groups [7] suggest that the transitions are distinct and their separation can be noticed with larger lattices.

In this paper we make a careful study of $SO(3)$ LGT (i.e. the $\beta=0$ line of the Creutz-Bhanot action) at $T\neq 0$, in order to arrive at a clear picture of the situation.The aim of our study is to get a clue to the nature of the high temperature phase. It is to be noted that the $SO(3)$ LGT does not have a global $Z_{2}$ symmetry of the corresponding $SU(2)$ LGT, which plays a crucial role [1, 2, 3] in the finite temperature transition. Also the finite temperature transition is of first order,if it is there at all. Nevertheles, we expect the two theories to have the same continuum limit. (We comment more on this later). It is therefore to be wondered whether the global $Z_{2}$ symmetry and universality arguments are relevant for the continuum theory. There are also other ways in which a study of $SO(3)$ LGT can yield us a clue to the nature of the high temperature phase. For instance, if the transition in $SO(3)$ LGT on symmetric lattices is really related to the finite temperature transition, we could conclude that the $Z_{2}$ monopoles are responsible for this effect of heating the system.

Most of the previous investigations have generally measured $\langle|L_{f}|\rangle$ instead of $\langle L_{f}\rangle$ to distinguish the high temperature phase.We have realized that it is far more instructive and not more difficult to measure $\langle L\rangle$ itself in order to understand the nature of the phases. It is also very useful to measure this order parameter in each Monte-Carlo sweep and follow its evolution. For the $SU(2)$ LGT in the high temperature phase $\langle L_{f}\rangle$ preferably settles to a positive value with a cold start. With a hot start it is equally likely to go to this state or the one with opposite sign of $\langle L_{f}\rangle$ [2]. Thus two distinct ground states related by the $Z_{2}$ transformation are explicitly seen. In contrast, in the low temperature phase, $\langle L_{f}\rangle$ is consistent with zero.

We have repeated a similar measurement for the adjoint Polyakov line $\langle L_{a}\rangle$ in $SO(3)$ LGT. There is no reason to regard this as an order parameter for the $SO(3)$ theory. $\langle L_{a}\rangle$ measures $\exp(-\beta F_{a})$ where $F_{a}$ is the free energy of a static quark in the adjoint representation of $SU(2)$. Even in the presumed confinement phase, such a quark can form a colour singlet bound state with the gluon, also in the adjoint representation, and thus give $F_{a}<\infty$. Nevertheless, we have found that $\langle L_{a}\rangle$ serves as a good observable to distinguish various phases.

We find that $\langle L_{a}\rangle$ is very small in the low temperature phase (Fig.2). It is, in fact, consistent with zero within errors. However there is no reason to expect it to be exactly zero, in contrast to $\langle L_{f}\rangle$ in the $SU(2)$ theory. In the strong coupling expansion, we get $\langle L_{a}\rangle=O(\beta_{a}^{4N_{\tau}})$ which is small for the couplings and lattice sizes we are using. This is presumably the reason for the smallness of $\langle L_{a}\rangle$. $\langle L_{a}\rangle$ has a far more interesting behaviour in the high temperature phase (Fig.2). With a cold start it settles to a positive value whereas with a hot start it settles to a negative value, distinct in magnitude (Fig.3). Thus it appears that there are two ground states for $SO(3)$ LGT at high temperatures, just as for the $SU(2)$ case. But, there is no indication of a $Z_{2}$ symmetry connecting the two ground states. We have measured the value of the free energy in the two ground states and find that they are equal within errors. Therefore it appears that there are two distinct and degenerate ground states.

We need to find the raison d’ etre of the two ground states. Note that

[TABLE]

and therefore have ranges $L_{f}=[-2,2]$ and $L_{a}=[-1,3]$, where $\exp\pm i(\theta/2)$ are the two eigenvalues of the $SU(2)$ matrix

[TABLE]

and $\theta$ has the range $[0,4\pi]$. It is to be noted that since U transforms under local gauge transformations as

[TABLE]

these eigenvalues are gauge invariant observables.

The $Z_{2}$ global symmetry group of the $SU(2)$ LGT corresponds to $\theta\rightarrow\theta+2\pi$. The fact that $\langle L_{f}\rangle$ is non zero and has either of two equal and opposite values can be interpreted as follows. The high temperature phase of $SU(2)$ LGT has configurations $\theta(mod4\pi)$ which are dominantly in one or the other sector, $[0,2\pi]$ or $[2\pi,4\pi]$. This bifurcation of the configurations is explicitly seen [2, 8]. In contrast, in the low temperature phase, the configurations are symmetrical [8] about $\langle\cos\theta/2\rangle=0$.

The variable $\langle L_{a}\rangle$ is insensitive to the $Z_{2}$ transformation. Therefore, if we were to probe the $SU(2)$ LGT using it, we would see just one ground state in the high temperature phase. Moreover this ground state would correspond to values in the region $[0,3]$ since $\langle\cos\theta\rangle$ is dominantly positive for such configurations.

This now provides an explanation for the ground state in the domain $[0,3]$ of the $SO(3)$ LGT at high temperatures, one that is preferred in the cold start. It is just the high temperature phase of the corresponding $SU(2)$ LGT. Histograms show (Fig.4a) that the configurations are peaked around positive values of $\cos\theta$ for this ground state.

We have to now account for the other ground state for which $\langle L_{a}\rangle$ is negative (Fig.4b). We now argue that this ground state is really the vaccuum of the large $\beta_{a}$ phase of the bulk transition in $SO(3)$ LGT. Strictly speaking the order parameters $\langle L_{f}\rangle$ and $\langle L_{a}\rangle$ are zero at $T=0$ because then the lattice extends indefinitely in the $x_{4}$ direction. However, we are working with lattices of finite extent in both the $x_{4}$ and spatial directions. Therefore $\langle L_{f}\rangle$ or $\langle L_{a}\rangle$ need not be strictly vanishing. In a finite lattice, phase transitions are impossible, in principle. At the most we can see metastable states and tunnelling between them. As we increase the lattice size, the system prefers to be in one of these metastable states longer and longer. In the thermodynamic limit, the system spends all the time in (one of equivalent) ground state(s).

The lattice of unequal extent in the $x_{4}$ and spatial directions can be interpreted equally as a finite volume version of a LGT with anisotropic couplings at $T=0$ or as a system at a non zero temperature. Therefore we would expect to find the metastable states corresponding to the large $\beta_{a}$ phase of the bulk transition in $SO(3)$ LGT and also the high temperature phase. Even though these states are not degenerate, with a skewed initial configuration the system may spend a long time in the state with a higher free energy. Indeed, the way to pin down the true ground state would be by using a hot start. Thus we would expect that on symmetric lattices, with a hot start the system settles to the vaccuum of the bulk transition. On the other hand, on an asymmetric lattice the hot start would choose the finite temperature ground state.

These expectations are borne out in our simulations and support the idea that the ground state with $\langle L_{a}\rangle$ in the interval $[0,3]$ corresponds to the high temperature phase and the one with $\langle L_{a}\rangle$ in the interval $[-1,0]$ corresponds to the bulk phase of $SO(3)$. In Fig.5 we notice that with a hot start, on the asymmetric lattice,the system settles into the phase with $\langle L_{a}\rangle$ positive. On the other hand, in case of a symmetric lattice the system prefers to be in the phase with $\langle L_{a}\rangle$ negative and small. On the large $8^{4}$ lattice this preference is even more marked [8]. We have found further evidence [8] to support this picture.

Our investigation confirms that the finite temperature transition is neither a lattice artifact nor an artifact of the action being used. $SO(3)$ LGT has a finite temperature transition analogous to that in the $SU(2)$ LGT, even though there is no global $Z_{2}$ symmetry to provide an order parameter in the strict sense. Instead, it is the bunching of configurations around a preferred domain of $\theta$ (the phase of the eigenvalues of the unitary matrix) which characterizes the high temperature phase. This is an important change in the characterization of this phase because it can be used in the presence of matter fields also. We use it elsewhere [9] to characterize the nature of the high temperature phase of QCD and to obtain a phenomenological model for it.

We feel the need to address two more questions to further justify the above conclusions. In the Bhanot-Creutz phase diagram (Fig.1) the large $\beta$ ($\beta_{a}$) phases of $SU(2)$ ($SO(3)$) theories are separated by a phase boundary. Therefore the continuum limits are taken in distinct phases. However there is another version of the mixed action by Halliday and Schwimmer [6] in which the large $\beta$ phases of $SU(2)$ and $SO(3)$ theories are continuously connected. Therefore the separation by a phase boundary in the Bhanot-Creutz model must be an artifact of the action chosen.

Even if we accept that the phase with $\langle L_{a}\rangle$ in the domain $[0,3]$ in the $SO(3)$ LGT is the high temperature phase, we have to tackle the problem that the transition is now of first order, in contrast to the second order transition in $SU(2)$ LGT. We need to reconcile this with its relevance to the continuum theory. We believe that the latent heat $\Delta L$ of this first order transition scales to zero in the continuum limit being considered. In other words, as we decrease the lattice spacing keeping the physical $\Lambda$ parameter fixed, $\Delta L/\Lambda$ scales to zero. We hope to check this in future simulations.

In this paper we have shown that simulations with finite size lattices may show metastable states related to the ground states of both bulk and finite temperature transitions and argued that the finite temperature transition persists independently of the specific action used even if the global symmetry is absent. We have suggested that a second order finite temperature transition can persist in the continuum theory even though it is of first order in $SO(3)$ LGT.