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Phase transitions in abelian lattice gauge theories


Phase transitions in abelian lattice gauge theories

Srinath Cheluvaraja? Theoretical Physics Group Tata Institute of Fundamental Research Homi Bhabha Road Mumbai - 400 005, INDIA

arXiv:hep-lat/9705041v2 15 Dec 1998

Abstract We study the phase transition in the abelian lattice gauge theory using the Wilson-Polyakov line as the order parameter. The Wilson-Polyakov line remains very small at strong coupling and becomes non-zero at weak coupling, signalling a con?nement to decon?nement phase transition. The decondensation of monopole loops is responsible for this phase transition. A ?nite size scaling analysis of the susceptibility of the Wilson line gives a ratio for γ/ν which is quite close to the corresponding value in the 3-d planar model. A scaling behaviour for the monopole loop distribution function is also established at the point of the second order phase transition. A measurement of the plaquette susceptibility at the transition point shows that it does not scale with the four dimensional volume as is expected of a ?rst order bulk transition.

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

?

e-mail:srinath@theory.tifr.res.in

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I. INTRODUCTION

Lattice gauge theories (LGTs) at non-zero temperature have been the focus of many investigations in recent years. Their study enables us to make non-perturbative predictions for the high temperature properties of gauge theories. Some of the issues which LGTs are expected to clarify are: the nature of the high temperature phase, the order of the phase transition, and the relevant elementary excitations at high temperatures. The pioneering work in [1] was the ?rst non-perturbative demonstration that quarks are decon?ned at high temperatures. This calculation is done in the strong coupling limit of the SU (2) LGT. Early Monte-Carlo simulations [2] provided further support for the existence of this phase transition. Since then, there have been many studies of the thermodynamic properties of the SU (2) and the SU (3) LGTs [3] which have provided valuable insights into the high temperature behaviour of gauge theories. In this paper, we study the properties of the U (1) LGT at non-zero temperature. More precisely, we study the U (1) LGT on asymmetric lattices for which Nσ >> Nτ . Nσ is the lattice size in the spatial direction and Nτ is the lattice size in the temporal direction. The limit Nσ → ∞ and Nτ ?xed, with periodic boundary conditions in the temporal direction, is the thermodynamic limit for a ?nite temperature system. There are several reasons why we have embarked on a study of this simple model. Firstly, the zero temperature properties of this model have been inferred from simulations on large symmetric lattices (Nσ = Nτ ) in [4]. In these simulations, a phase transition is observed from a con?ning phase to a decon?ning phase. Though there was some controversy about the order of the phase transition, recent simulations on large lattices strongly suggest that the bulk transition is of ?rst order [5]. The mechanism of this bulk transition is quite well understood in terms of monopole excitations [11]. These monopoles are topological objects that arise because of the periodicity properties of the action. The bulk system exists in two phases, a con?ning phase in which the monopole currents condense causing complete Meissner e?ect, and a decon?ning phase in which the monopoles are too heavy to have any physical e?ect. It would be interesting to see how this picture of con?nement vs decon?nement gets a?ected at ?nite temperature. This is also of relevance from the point of view of some recent claims of there being a second order phase transition in an extended version of the U (1) LGT with a monopole chemical potential term [18]. A de?nition of a continuum theory is possible at the point of the second order phase transition, and an investigation of its properties can be 2

quite instructive. Secondly, the critical behaviour of gauge theories at high temperatures can be understood from analogous behaviour of three-dimensional spin models. The strong coupling analysis in [1] shows that the partition function of the SU (2) LGT can be rewritten as a three-dimensional spin model with a global Z(2) symmetry. The decon?ning phase corresponds to the symmetry breaking phase of the spin model, and the con?ning phase corresponds to the symmetric phase. This symmetry arises because ?nite temperature gauge systems have an additional global symmetry arising from the periodic boundary conditions in the temporal direction. For systems with a gauge symmetry group G, this global symmetry group consists of elements belonging to the center of the gauge group. Furthermore, as emphasized in [7], the order of the phase transition in four dimensions is expected be dictated by the universality classes present in 3-d spin models having this global symmetry. These expectations have been borne out for the SU (2) [8] and the SU (3) LGTs [9] in which one observes a second order Ising like and a ?rst order Z(3) like phase transition respectively. One of our aims is to examine the same issue in the much simpler U (1) LGT. Unlike the non-abelian SU (N ) LGTs, which have a discrete center subgroup, (Z(N )), the abelian U (1) LGT has a continuous center subgroup which is identical to the group itself. The role of the U (1) group on the con?nement to decon?nement transition is also of some interest from the point of view of the abelian dominance hypothesis for con?nement which holds that a U (1) subgroup controls the non-perturbative dynamics of non-abelian gauge theories [19]. Finally, as the U (1) LGT already has a bulk phase transition unlike the SU (2) and SU (3) LGTs, there is the question of the interplay between this transition with the expected ?nite temperature transition. This matter has been recently examined in the context of SU (2) LGTs using a mixed action; it was found in [10] that the bulk transition and the ?nite temperature transition may coincide, making it di?cult to distinguish one from the other. Since simulations are always done on ?nite lattices, this raises the question whether the earlier studies on symmetric lattices [24] were seeing a bulk transition or a ?nite temperature transition. As argued in [10], since ”zero temperature” properties are inferred by extrapolating the results on symmetric lattices, and ”?nite temperature” properties are inferred by extrapolating the results on asymmetric lattices, it is not very easy to decide whether one is seeing a bulk e?ect or a genuine ?nite temperature e?ect. A major di?erence between a bulk transition and a ?nite temperature transition, which

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can in principle distinguish the two transitions, is the movement of the transition point with the temporal lattice size. For non-abelian gauge theories, a physically relevant ?nite temperature transition should move towards the weak coupling region in a manner which is speci?ed by the beta function of an asymptotically free gauge theory. With these notions in mind, we would like to investigate the ?nite temperature properties of abelian lattice gauge theories. We will mainly consider the Wilson action [20] for the U (1) LGT, which is given by S=β
n?>ν

cos(θ(n ?ν)).

(1)

The θ(n ?ν) are the usual oriented plaquette variables: θ(n ?ν) = θ(n ?) + θ(n + ? ν) ? θ(n + ν ?) ? θ(n ν). (2)

The link variables θ(n ?) can take values from ?π to π. As mentioned before, the properties of this model at zero temperature are well known. There is a transition at β ≈ 1.0 that is caused by a decondensation of monopole currents ( the bulk transition on a 164 lattices has been located precisely in [23] and occurs at β = 1.016). These monopole currents are de?ned on the dual lattice by counting the number of Dirac strings entering or leaving a three dimensional cube on the original lattice [21]. The monopole density on a link (?l) of the dual lattice is de?ned as ρ(?l) = ? θ(p) is extracted from θ(p) by expressing it as ? θ(p) = θ(p) + 2πn(p), (4) ?1 2π ? θ(p);
p∈c

(3)

so that it takes values from ?π to π. In the above expression, the monopole current is de?ned on the links ?l of the dual lattice which are dual to the cubes c in the original lattice. As the monopoles form closed loops on the dual lattice, it is more convenient to measure the total perimeter density of the monopole current loops. The perimeter density is given by ρ= 1 Nl ρ(l) ;
l

(5)

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Nl is the total number of links in the lattice. One observable which is of special relevance to our analysis is the Wilson-Polyakov line (henceforth called the Wilson line), which is de?ned as


L(n) =
n0 =0

exp(iθ(n + n0 ? ? 4 4)).

(6)

This observable is gauge invariant and has played a crucial role in studies of the decon?nement transition in SU (N ) LGTs. The center symmetry that emerges at ?nite temperature is the transformation which multiplies by a constant phase all the time-like links emanating from some ?xed time slice, namely 4)). 4)) exp(iθ(n ? → exp(i α) exp(i θ(n ? Although the action is invariant under this transformation, the Wilson line transforms as: L(n) → exp(i α)L(n). (8) (7)

Hence, a non-zero expectation value of the Wilson line signals a spontaneous breakdown of the center symmetry. The Wilson line is given the usual physical interpretation by writing it in the form: L(n) = exp(?βFq (n)). (9)

It measures the free energy (Fq (n)) of a static charge in a heat bath at a temperature β ?1 . Hence, a nonzero value of L(n) indicates decon?nement of static charges whereas a zero value indicates con?nement. We point out a subtlety in this representation, which has also been noticed before in the context of the SU (2) LGT [15]. The left hand side is in general a complex quantity and hence cannot be given a freeenergy interpretation. One way of avoiding this problem is to work with the correlation function of two Wilson lines which is always a non-negative quantity. Apart from the Wilson line, we have also studied the susceptibility of the Wilson line which is de?ned as
3 χl = Nσ ( s2 ? |s| 2 ). .

(10)

The ”spin” variable s is constructed in Eq. 14. The peak in the Wilson line susceptibility can be used to locate the phase transition. A related observable that is also relevant is the plaquette susceptibility, de?ned as

5

3 χp = 6Nσ Nτ p

( P2 ? P )

2

.

(11)

In the above equation, P is the average plaquette density in the system. The peak in the plaquette susceptibility can also be used to locate the phase transition. According to the ?nite size scaling theory, the susceptibility of an observable, which blows up at a phase transition transition point in the in?nite volume limit, is expected to scale with the system size as χcr = N α p ; (12)

here N is the size of the system. For a ?rst order transition, α equals d, the dimensionality of the system ; for a second order transition, α equals γ/ν, where γ and ν are the exponents for the susceptibility and the correlation length near the transition, which are given by χ = (T ? Tc )?γ ξ = (T ? Tc )?ν . (13)

We will use the above observables to study the phase transition in the abelian lattice gauge theory. This paper is organized as follows. Sec 2 contains the results of our numerical investigations of this model. In Sec 3 we make some comments on the mixed action U (1) LGT and compare and contrast it with that of the of mixed action SU (2) LGT. In Sec 4 we summarize our conclusions.

II. THE U (1) LGT AT NON-ZERO TEMPERATURE.

In this section we present our numerical analysis of the model. The system is mimicked at ?nite temperature by working on an asymmetric lattice (Nσ >> Nτ ) with periodic boundary conditions in the Euclidean time direction. We ?rst brie?y describe the numerical procedure that we adopted to obtain our results. The Metropolis algorithm was used to generate successive Monte-Carlo con?gurations. A new link variable θ′ was generated from the old one θ by adding a number randomly chosen in the range (?α, α) with uniform probability. The value of α was tuned to get an acceptance of 50 percent. Care was taken so that the link variables remained in the range (?π, π). Simulations were performed on temporal lattices with Nτ = 2, 3 and 4. The observables that were measured are: the monopole density, the Wilson line, the plaquette density, the Wilson line susceptibility, and the plaquette susceptibility. As the Wilson line is

6

complex ( it is a phase with modulus equal to one ), we measure its real and imaginary parts separately. If we simply measure the average value of the real or the imaginary parts, the result will always be equal to zero because phase transitions are impossible on ?nite system as the tunnelling between the degenerate states always restores the symmetry. A rigorous way of studying symmetry breaking is to study the average value of the Wilson line in the presence of a small symmetry breaking external ?eld, and then take the limit of zero ?eld after taking the large volume limit. A simpler prescription, that is often employed in studying continuous spin models, is to study the root mean square value of the order parameter, and this is how we will proceed. The observable that we have measured as an indicator of spontaneous symmetry breaking is s ; s is de?ned as s = ReL2 + ImL2 . (14)

Here ReL and ImL refer to the average of the real and imaginary parts of the Wilson line respectively. A simple strong coupling analysis (valid for β small) yields the following e?ective action for the Wilson lines: β Nτ Sef f = 2( ) 2 cos(θ(n) ? θ(n′ )) ;
nn′

(15)

Nτ is the temporal extent of the lattice and the θ(n) variables are the sums of the phases of all the time like links at the spatial point n. This is the action for the three dimensional planar model which is known to have an order-disorder transition at βcr = 0.454. For an Nτ = 2 lattice, this gives the critical coupling to be approximately 0.95. Thus we expect our lattice model to have a phase transition at β ≈ 0.9. This strong coupling argument is valid only if the phase transition takes place in the strong coupling regime. Nevertheless, the strong coupling approximation provides a simple way of seeing how a three-dimensional spin model emerges from the four dimensional gauge theory. A more direct way of seeing the appearance of an e?ective 3-d spin model, without using a strong coupling approximation, is to use the dual representation of the 4-d U (1) LGT. The dual representation is given by Z =
m? (r)

exp ?

1 2β

(?? φν ? ?ν φ? )2 + 2πi
n ?ν n

m? (n)φ? (n)

(16)

This describes a gas of closed monopole loops (m? (r)) which interact by a 4-dim Coulomb potential. This 7

system has a phase transition which takes place as a result of the competition between the energy and the entropy of the monopole loops. The ?elds φ? (r) can be integrated out to give Z =
m? (r)

exp(?2π 2 β
n n′

m? (n)G? ν (n ? n′ )mν (n′ ))

(17)

At non-zero temperature, the time direction is ?nite, and the Green’s function satisfy periodic boundary conditions. The four dimensional Green’s function can be rewritten as: ? G? ν (r ? r′ ) ≈ T G? ν (r ? r′ ) ; (18)

? here G? ν (r ? r′ ) is the 3-dimensional Green’s function. As the gas is ?nite in one direction, one again has a gas of monopole loops which now e?ectively interact with a 3-dim Coulomb interaction. From the point of view of the e?ective three-dimensional planar model, the monopole loops behave like vortex lines. The entropy of large loops in three-dimensions is smaller than in the four-dimensional case and this shifts the transition. The order of the transition cannot be deduced from these energy arguments and has to be determined by doing a ?nite size scaling analysis. We show in Fig. 1 the variation of a 63 2 lattice. The observable s with β on

s(n) is close to zero at small β and rises smoothly across the critical

value. The U (1) monopole density variation is shown in Fig. 3. There is a fall in the monopole density across the transition which coincides with the rise in the order parameter. In both cases, the variation is quite gradual and we would suspect that we are in the vicinity of a second order transition. This is made further suggestive by the gradual rise in the plaquette expectation value (see Fig. 2). To determine the order of the transition, we perform a ?nite size scaling analysis of the susceptibility of the order parameter (χl ). We have done the ?nite size scaling study on lattices of temporal extent Nτ = 3 and Nτ = 4. For this method to work, it is crucial that we are very close to the pseudo-critical point corresponding to the lattice that we are working on. The histogram method is used to extrapolate observables from one value to a nearby neighbouring value. The pseudo-critical point is located in this way by looking at the peak in the susceptibility of the order parameter. The behaviour of the susceptibility near the transition on 6, 9, 12 and 16 sized spatial lattices (keeping the temporal extent ?xed at Nτ = 3) is shown in Fig. 4. These results were got after 200,000 measurements on Nσ = 6, 9 lattices while 150,000 measurements were made on the Nσ = 12 lattice and 100,000 measurements were made on the Nσ = 16 lattice. All the measurements were 8

made after ignoring the ?rst 50,000 Monte-Carlo iterations. The errors were estimated by binning the data. The ?nite size scaling theory predicts
d χ ≈ Nσ

(19)

for a ?rst order phase transition; d being the e?ective dimension of the system (in this case d = 3); for a second order transition, the prediction is
γ ν χ ≈ Nσ ;

(20)

α γ and ν are the exponents in Eq. 13. The peaks in the susceptiblity were ?t to an Nσ dependence and

a good ?t is obtained (Fig. 5). From our ?t we determine γ/ν to be 1.95 with an error of 0.13. For the three dimensional planar model, which is the e?ective spin model with which we would like to compare our results, the ratio γ/ν = 1.97. We have performed another test for the order of the phase transition by studying the distribution of the monopole loops. Since the monopole loops, which now behave as vortex lines, are responsible for driving the phase transition, they should also exhibit a scaling behaviour at the point where the transition becomes second order. At the point of the second order phase transition, there are ?uctuations on all length scales and the monopole loops come in all sizes and shapes. At the transition point, the monopole loop distribution function, p(l), should scale as p(l) = 1 lτ (21)

with an exponent τ which is independent of the volume. The function p(l) is de?ned as the probability of ?nding a loop of length l at a site. We have calculated p(l) for various lattices at their pseudo-critical points and we ?nd that p(l) can be nicely ?tted to the above form. The value of τ that we get (using data on a Nσ = 16 lattice) is 2.27 ± .003. Unfortunately, we are not able to compare it with a theoretical calculation. Nevertheless, it is interesting that the topological excitations, in this case the monopoles, exhibit a scaling dependence at the transition point. Fig. 6 shows p(l) for loops of length l upto 14 on di?erent lattices. The plot shows that they fall on a straight line independent of the volume. The slight distortions for the Nσ = 6 lattice are due to ?nite size corrections when the loop length is larger than the 9

lattice size. A study of larger loop lengths can also be made but the algorithm that we have used to count the loops slows down when the loop lengths become very large, and so we have not studied larger loops. Away from the critical point, p(l) either falls rapidly to zero (in the decon?ning phase) or remains constant (in the con?ning phase). A similar study of the monopole loop distribution function was made in [22] for the abelian projected monopoles in the SU (2) theory near the continuum limit. Now we say a few words on the location of this transition as compared to the location of the bulk transition. The transition point (which is located by looking at the peak in the susceptibility of the order parameter) shifts as a function of Nτ as: Nτ βcr 2 0.9297(3) 3 1.012(2) 4 1.032(2)

The above critical values are those obtained on Nσ of 16, 16 and 12 respectively. The bulk transition on a 164 lattice was located at β = 1.016 in [5]. Since the bulk transition is known to be of ?rst order [5], and the transition that we have observed is of second order, we are observing a change in the order of the phase transition as a function of Nτ . The nature of the transition that we have observed can be further studied by monitoring the behaviour of the plaquette susceptibility which was de?ned in Eq. 11. The behaviour of the plaquette susceptibility for a 63 3 and a 123 3 lattice is shown in Fig. 8. This graph clearly shows that the plaquette susceptibility does not scale with the four dimensional volume as is expected of a ?rst order bulk transition. The above observations show that the transition that we have observed (on an Nτ = 3 lattice) is a decon?nement transition whose scaling behaviour is quite distinct from that of the bulk transition. The order of the transition as seen by the Wilson line is a second order transition with the ratio γ/ν which has approximately the same value as that in the three-dimensional planar model. Since the plaquette susceptibility does not scale as a ?rst order bulk transition, we are not observing the bulk transition at the point where we are observing the ?nite temperature transition. The ?nite size scaling analysis can be repeated on an Nτ = 4 lattice and the peak in the susceptibility of the order parameter as a function of volume is shown in Fig. 7. This scaling again suggests a second order transition with a value for γ/ν which is again in good agreement with that of the three-dimensional planar model.

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We now place our results in the context of some recent developments. The studies in [10] considered the ?nite temperature properties of the mixed action SU (2) LGT which is de?ned by S = (βf /2)
p

trf U (p) + (βa /3)
p

tra U (p).

(22)

This model is known to have lines of ?rst order bulk transitions in the βf , βa plane [24]. It was found in [10] that the decon?nement transition of the pure SU (2) LGT, which is of second order, continues into the βf ,βa plane and joins the line of ?rst order bulk transitions. Based on this observation in [10], a possibility was considered where the there is either only a bulk or a ?nite temperature transition in the SU (2) LGT. On the other hand, there may also be a very small but ?nite separation in the two transitions which cannot be resolved on the lattice sizes used in the simulations. In our case, the relevant coupling space is the β, T plane. Our simulations of the U (1) LGT show that there is a decon?nement transition which is of second order and that the plaquette susceptibility does not scale as a ?rst order bulk transition at the transition point. The transition is shown to shift from its bulk value as a function of Nτ but this shift is very small, though still discernible. We also place our results in the light of the expectations of Svetitsky and Ya?e [7]. According to their general arguments, the critical begaviour of the U (1) LGT at ?nite temperature is expected to fall in the same universality class as that of the 3-d planar model, which is known to have a second order phase transition. Our ?nite size scaling analysis on the Nτ = 3 lattice de?nitely rules out a ?rst order phase transition and indicates that the ratio γ/ν has approximately the same value as in the 3-d planar model. We conclude this section with a proposal for the ?nite temperature phase diagram for the U(1) LGT. In the bulk system (which corresponds to taking the limit Nσ = Nτ and Nσ → ∞), there is the monopole driven phase transition. On asymmetric lattices of very small temporal extent, we expect the bulk system to look like a three dimensional U (1) gauge theory. The three-dimensional U (1) LGT has no bulk transition. On asymmetric lattices of very large temporal extent we expect to see the bulk transition. Lattices of intermediate temporal extent will exhibit a complicated crossover from a four-dimensional bulk system to a three-dimensional bulk system. Our simulations on an Nτ = 3 lattice show that there is a second order phase transition which has the same value of γ/ν as in the three-dimensional planar model. This transition is a decon?nement transition as is indicated by the behaviour of the fundamental Wilson line. However, 11

there is no bulk transition on this lattice.

III. MIXED ACTION U (1) LGT.

Since lattice actions are anyway not unique, we can always construct more complicated looking lattice actions and examine their properties. A simple generalization of the action in Sec 1 is the mixed action which is de?ned by S = β1
p

cos(θ(p)) + β2
p

cos(2θ(p)).

(23)

The two pieces of the above action are di?erent from each other only insofar as their periodicity properties are concerned. In the naive continuum limit, a → 0, the second term is like an irrelevant coupling and is not expected to change the long distance properties of the theory. The zero temperature properties of this action have been studied in [17] and it has a rich phase structure of ?rst and second order transitions. This system can also be studied at ?nite temperature just as we did for the β2 = 0 theory. Again, from a simple strong coupling analysis of the mixed action U (1) LGT, we get an e?ective theory of spins which is that of the mixed planar model Sef f = 2( β1 N τ ) 2 cos(θ(n) ? θ(n′ )) + 2(
nn′

β2 N τ ) 2

cos(2θ(n) ? 2θ(n′ )).
nn′

(24)

Putting β1 = 0 gives a three-dimensional planar model of spins, the only di?erence being in the periodicity properties of the action. Hence, the ?nite temperature properties of the mixed model in the β1 = 0 limit should be identical to those in the β2 = 0 limit. In particular, the previous statements regarding the order of the transition will also be true in this limit. A surprising feature of the mixed planar model is that it posesses a region of ?rst order phase transitions for some values of β2 [26]. This implies a similar region of ?rst order transitions in the mixed U (1) LGT for a segment of β2 values. The order of the ?nite temperature transition changing in the direction of an irrelevant coupling has also been discussed in the context of the mixed action SU (2) LGT [10]. From the above analysis, it is clear that there are many similarities between the U (1) LGT and the mixed action SU (2) LGT. We now show that the similarity also extends to the construction of the relevant order parameters in the two models. The mixed action SU (2) LGT [24] is de?ned by 12

S=

βf 2

trf U (p) +
p

βa 3

tra U (p);
p

(25)

trf and tra denote the traces in the fundamental and the adjoint representations respectively. The limit βa = 0 describes an SU (2) LGT and the limit βf = 0 describes an SO(3) LGT. The order parameter of the ?nite temperature transition in the SU (2) LGT is the Wilson line in the fundamental representation which is de?ned as


Lf (n) = T rf
n0 =0

U (n + n0 ? ? 4 4).

(26)

In the SO(3) LGT (βf = 0), this observable is identically zero because of the following local Z(2) symmetry: U (n ? → Z(n)U (n ? 4) 4); (27)

Z(n) can take the values +1 or ?1 at any site. For the SO(3) LGT, the appropriate order parameter is the Wilson line in the adjoint representation,


La (n) = T ra
n0 =0

U (n + n0 ? ? , 4 4)

(28)

which is invariant under the local Z(2) transformation in Eq. 27. For the group SU (2), Lf and La are related by La (n) = Lf (n) ? 1.0 .
2

(29)

In the mixed action U (1) LGT we are faced with a similar problem in de?ning the order parameter for the β1 = 0 theory. In this limit, the Wilson line de?ned in Sec 1 is identically zero because of the following local symmetry: exp(iθ(n ? → Z(n) exp(i θ(n ? 4)) 4)). The correct order parameter to use in this limit is


(30)

L2 (n) =
n0 =0

4 4)) , exp(i 2θ(n + n0 ? ?

(31)

which is analogous to the Wilson line in the adjoint representation of SU (2). The relationship between L2 (n) and L(n) is L2 (n) = L(n) . 13
2

Though there are similarities at the formal level, and in the phase structure of the mixed action abelian and mixed action non-abelian theories, there are, of-course, many important di?erences between the two systems. An important di?erence is that, unlike in the abelian LGT, the critical coupling in the SU (2) LGT is expected to scale with Nτ according to the beta function of the Yang-Mills theory. There is strong evidence for this asymptotic scaling from simulations on very large temporal lattices [25]. The purpose of this section was only to indicate that many of the issues such as the mixing of the bulk and ?nite temperature transitions which have been raised recently can all be explored in the much simpler mixed action abelian lattice gauge theory. Also, because the physical properties of this model are well understood in terms of the monopole excitations, this model may prove useful in investigating these issues.

IV. CONCLUSIONS

In this paper we studied the U (1) LGT using the Wilson line as the order parameter. We have found that there is a transition into a decon?ning phase at large coupling which is driven by the decondensation of monopole loops. The monopole loops, however, e?ectively interact like the vortices in the three-dimensional planar model. The decon?ning phase breaks the global U (1) symmetry present in the theory. A ?nite size scaling analysis of the susceptibility of the Wilson line indicates that the transition is of second order ( on lattices of temporal size Nτ = 3 and Nτ = 4) with a ratio for γ/ν which has almost the same value as in the three-dimensional planar model. A scaling form for the distribution of the monopole loops was also established at the point of the second order phase transition. This transition was also examined by studying the plaquette susceptibility at the transition point. The plaquette susceptibility (on a Nτ = 3 lattice) does not scale as is expected of a ?rst order bulk transition. There is also a small shift in the transition point from the bulk value. We have also pointed out that many of the recent issues concerning the mixing of the bulk and ?nite temperature transitions can also be raised in the abelian lattice gauge theory. Since the abelian theories are well understood in terms of their monopole excitations, some of these issues can perhaps be clari?ed.

14

[1] A. Polyakov, Phys. Lett. 72B, 477 (1978) ; L. Susskind, Phys. Rev. D20, 2610 (1978). [2] J. Kuti, J. Polonyi, and K. Szlachanyi, Phys. Letters. B98, 1980 (199); L. McLerran and B. Svetitsky, Phys. Letters. B98, 1980 (195). [3] J. Engels, F. Karsch, H. Satz and I. Montvay, Phys. Lett. B101, 89 (1981); J. Engels, F. Karsch, H. Satz and I. Montvay, Nucl. Phys. B205, 545 (1982); H. Satz, Nucl. Phys. 252, 183 (1985); J. Engels, J. Fingberg, F. Karsch, D. Miller, and M. Weber, Phys. Lett. B252, 625 (1990); J. Engels, F. Karsch and K. Redlich, Nucl. Phys. B435, 295 (1995); C. Boyd, J. Engels, F. Karsch, E. Laermann, C. Legeland, M. Lutgemeier, and B. Petersson, Nucl. Phys. B469, 419 (1996). [4] M. Creutz, L. Jacobs and C. Rebbi, Phys. Rev. D20, 1915 (1979);B. Lautrup and M. Nauenberg, Phys. Lett. 95B, 63 (1980); G. Bhanot, Phys. Rev. D24, 461 (1981); K. Moriarty, Phys. Rev. D25, 2185 (1982); D. Caldi, Nucl. Phys. 220[FS8], 48 (1983). [5] J. Jersak and B. Lang, Nucl. Phys. B251, 279 (1985). [6] K. G. Wilson, Phys. Rev. D10, 2445 (1974). [7] L. Ya?e and B. Svetitsky, Nucl. Phys. B210[FS6], 423 (1982); L. Mclerran and B. Svetitsky, Phys. Rev. D26, 963 (1982); L Mclerran and B. Svetitsky, Phys. Rev. D24, 450 (1981); B. Svetitsky, Phys. Rep. 132, 1 (1986). [8] R. Gavai, H. Satz, Phys. Lett. B145, 248 (1984); J. Engels, J. Jersak, K. Kanaya, E. Laermann, C. B. Lang, T. Neuhaus, and H. Satz, Nucl.Phys. B280, 577 (1987); J. Engels, J. Fingberg, and M. Weber, Nucl. Phys. B332, 737 (1990) ; J. Engels, J. Fingberg and D. Miller, Nucl. Phys. B387, 501 (1992) ; [9] K. Kajantie, C. Montonen, and E.Pietarinen, Z. Phys. C9, 253 (1981); T. Celik, J. Engels, and H. Satz, Phys. Lett. 125B, 411 (1983); J. Kogut, H. Matsuoka, M. Stone,H. W. Wyld,S. Shenker,J. Shigemitsu, and D. K. Sinclair, Phys. Rev. Lett 51, 869 (1983); J. Kogut, J. Polonyi, H. W. Wyld, J. Shigemitsu, and D. K. Sinclair,Nucl.Phys. B251, 318 (1985). [10] R. Gavai, M. Mathur and M. Grady, Nucl. Phys. B423, 123 (1994); R. V. Gavai and M. Mathur, B448, 399 (1995). [11] T. Banks,J. Kogut and R. Myerson, Nucl. Phys. 129,493 (1977). [12] I. G. Halliday and A. Schwimmer, Phys. Lett. B101, 327 (1981); I. G. Halliday and A. Schwimmer, Phys. Lett. B102, 337 (1981); R. C. Brower, H. Levine and D. Kessler, Nucl. Phys. B205[FS5], 77 (1982). [13] G. Bhanot and M. Creutz, Phys. Rev. D24, 3212 (1981). [14] S. Elitzur, Phys. Rev. D12, 3978 (1975).

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[15] J. Kiskis, Phys. Rev. D51, 3781 (1992). [16] P. H. Daamgard, Phys. Lett. B183, 81 (1987); P. H. Daamgard, Phys. Lett. B194, 107 (1987); J. Fingberg, D. Miller, K. Redlich, J. Seixas, and M. Weber, Phys. Lett. B248, 347 (1990). [17] G. Bhanot, Nucl. Phys. B205[FS5], 168, (1982). [18] W. Kerler,C. Rebbi and A. Weber, Nucl. Phys. B (Proc. Suppl), 53, 503 (1997). [19] G. ’t Hooft, Nucl. Phys B190, 455 (1981). [20] K. G. Wilson, Phys. Rev. D10, 2445 (1974). [21] T. DeGrand and D. Touissant, Phys. Rev. D 22, 2478 (1981). [22] M. Grady, hep-lat 9802035 [23] J. Jersak, T. Neuhaus and P. Zerwar, Phys. Lett B133, 103 (1983). [24] G. Bhanot and M. Creutz, Phys. Rev. D24, 3212 (1981). [25] J. Fingberg, U. Heller, and F. Karsch, Nucl. Phys.392, 493 (1993). [26] W. Janke and H. Kleinert, Nucl. Phys. B270, 399 (1986).

16

FIG. 1. Variation of the order parameter on a 63 2 lattice.

FIG. 2. Plaquette expectation value on a 63 2 lattice.

FIG. 3. Monopole density on a 63 2 lattice.

FIG. 4. Susceptibility of the order parameter near the transition on 6, 9, 12 and 16 size spatial lattices.

FIG. 5. A ?t of the maximum value of the logarithm of the susceptibility to lnN .

FIG. 6. Scaling of the monopole loop distribution function p(l) with loop length (l). The graph shows the logarithm of p(l) as a function of the logarithm of l.

FIG. 7. A ?nite size study of the susceptibility of the order parameter on a Nτ = 4 lattice.

FIG. 8. lattices.

The scaling of the plaquette susceptibility near the phase transition. This is shown on 63 3 and 123 3

17

1 0.9 0.8 0.7

ps

3 3 3 33
0.6 0.8 1

3 3

3

3

0.6 0.5 0.4 0.3 0.2 0.1 0.2

3
0.4

3

1.2

1.4

1.6

1.8

2

1 0.9 0.8 0.7 0.6 cos (p)=2 0.5 0.4 0.3 0.2 0.1 0.2

33 3 3
0.4

3

3

3 3

3

3

3
0.6 0.8 1 1.2 1.4 1.6 1.8 2

0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0.2 0.4 0.6 0.8 1

3

3 33 3 3 3 3 3
1.2 1.4 1.6

3
1.8

3
2

30 "6" "9" "12" "16" 25

20

chi

15

10

5

0 0.975

0.98

0.985

0.99

0.995

1

1.005 beta

1.01

1.015

1.02

1.025

1.03

4

3.5

3

3 3 3 3
1.8 2 2.2 2.4 2.6 2.8 3

log max

2.5

2

1.5

1

0.5

logN

-3.5 "6" "9" "12" "16"

-4

-4.5

-5 log p(l) -5.5 -6 -6.5 -7 1.2

1.4

1.6

1.8

2 log(l)

2.2

2.4

2.6

2.8

14 "6" "8" "12" 12

10

8 chi 6 4 2 0 1.015

1.02

1.025 beta

1.03

1.035

5 "6" "12" 4.5

4

chi

3.5

3

2.5

2 1 1.005 1.01 beta 1.015 1.02 1.025


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