The phase structure of lattice Yang-Mills theories

IL NUOVO CIMENTO VOL. 104 B, N. 2 Agosto 1989

The Phase Structure of Lattice Yang-Mills Theories (*).

A. PATRASCIOIU(**) and E. SEILER Max-Planck-Institut fi~r Physik und Astrophysik Werner-Heisenberg-Institut f~r Physik P.O. Box 401212, Mi~nchen, BRD

V. LINKE and I. O. STAMATESCU(***)

Institut fi~r Theorie der Elementarteilchen, Freie Universitdt Berlin Arnimallee 14, D-IO00 Berlin 33

(ricevuto il 15 Maggio 1989)

Summary. - - A recently proposed general mechanism for the occurrence of phase transitions is investigated in the context of lattice gauge theories. It leads to the prediction that all zero-temperature lattice gauge theories in D~>3 must undergo a phase transition; it is the limit of the finite temperature deconfining transition. Numerical data corroborating our assertion are presented for the 4D SU(2) lattice gauge theory.

PACS 11.10 - Field theory. PACS 02.10 - Logic, set theory and algebra.

In a recent paper (1) we proposed a general mechanism for the occurrence of phase transitions in a class of models including ferromagnets with nearest- neighbour interaction and lattice Yang-Mills theories. I t is based on a simple energy-en t ropy est imate for the ,(defects), (domain walls or vor tex networks) of

(*) To speed up publication, the authors of this paper have agreed to not receive the proofs for correction. (**) Permanent address: University of Arizona, Physics Dept., Tucson, AZ 85721, USA. (***) Present address: FEST, Schmeilweg 5, D-6900 Heidelberg, BRD. (1) A. PATRASCIOIU, E. SEILER and I. O. STAMATESCU: From ice to deconfinement: a unifying view of certain phase transitions, MPI-PAE-PTh 75/88.

229

236 A. PATRASCIOIU, E. SEILER, V. LINKE and I. O. STAMATESCU

the dual model. When tested against numerically or analytically known phase transitions, it proves very successful in predicting their location. It also predicts some novel transitions(Z), some of which according to conventional wisdom should not exist, for instance in the 2D O(N) models (N > 2) (3). Furthermore, it predicts phase transitions in all lattice gauge models for D > 2. We supported this prediction by a numerical study of the 3D U(1) (compact) lattice gauge theory, where we found strong evidence for a first-order transition at the predicted value of the coupling(4). A physically interesting prediction of our energy-entropy argument is that there must be a phase transition also in all non- Abelian lattice gauge models in 4D (at zero temperature). Of course it is generally accepted that many zero-temperature lattice gauge models have first- order transitions: the SU(N) ( N > 3) and the U(1) model with Wilson's action in 4D, the SU(2) model with Wilson's action in 5D, the SU(2) model with a next- nearest-neighbor interaction in 4D and the SU(2) model with an action containing both the fundamental and the adjoint representations in 4D (for a comprehensive review see (~)). It is also a well-known and rigorously proven fact that all finite-temperature lattice gauge models have a deconfining transition in D I> 3 (~.7). Our considerations suggest that all these transitions are driven by the same mechanism and that the deconfining transition at nonzero temperature smoothly joins the zero temperature (,,bulked) transition, for all gauge groups considered. In fact, it has been noticed that numerically (for N i> 4 in 4D) it is difficult, if not impossible, to separate the zero-temperature limit of the deconfining transition from the bulk one(8). Even for SU(2) and SU(3), the numerical data for the deconfining transition (~12) extrapolate to a value at zero

(2) A. PATRASCIOIU: Phys. Rev. Lett., 58, 2285 (1987). (s) E. SEIZER, I. O. STAMATESCU, A. PATRASCIOIU and V. LINKE: Critical behavior, scaling and universality in some two-dimensional spin models, MPI-PAE/PTh 76/88, to appear in Nucl. Phys. B. (4) E. SEIZER: Challenging the conventional wisdom on ferromagnets and lattice gauge theories, MPI-PAE/PTh 24/88, to appear in Proceedings of the 1988 Karpacz Winter School. (5) M. CREUTZ, L. JACOBS and C. REBBI: Phys. Rev., 95, 201 (1983). (~) C. BORGS and E. SEIZER: Nucl. Phys. B, 215 [FS7], 125 (1983); Commun. Math. Phys., 91, 329 (1983). (7) E. T. TOMBOULIS and L. YAFFE: Commun. Math. Phys., 100, 313 (1985). (8) S. DAS and J. KOGUT: Phys. Lett. B, 141, 105 (1984); Nucl. Phys. B, 257 [FS14], 141 (1985); G. BATROUNI and B. SVETITSKY: Phys. Rev. Lett., 152, 2205 (1984); K. FABRICIUS, O. HAAN and F. KLINKHAMER: Phys. Rev. D, 30, 2227 (1984). (9) j . ENGELS, J. JERS.~K, K. KANAYA, E. LAERMANN, C. B. LANG, W. NEUHAUS and H. SATZ: Nucl. Phys. B, 280, 577 (1987). (lo) A. D. KENNEDY, J. KUTI, S. MEYER and B. J. PENDZETON: Phys. Rev. Lett., 54, 87 (1985). (11) N. CHRIST and E. TERRANO: Phys. Rev. Lett., 56, 111 (1986). (,2) S. A. GOTTZIEB, J. KUTI, D. TOUSSAINT and A. D. KENNEDY: Phys. Rev. Lett., 55, 1958 (1985).

THE PHASE STRUCTURE OF LATTICE YANG-MILLS THEORIES 231

temperature that is in the range predicted by our energy-entropy argument. In this paper we will argue that also for SU(2) and SU(3) there is a first-order deconfining bulk transition which is the zero-temperature limit of the well- known deconfining transition at positive temperature; we corroborate this statement by a numerical study of the SU(2) lattice gauge theory.

Let us review the physical picture we are proposing for these phase transitions. At strong coupling a lattice gauge theory with compact gauge group G can be described as a dilute gas of certain polymers (which may be considered as the defect networks of the dual system) as follows: the Gibbs factor for each plaquette p can be expanded into a series involving the characters of the irreducible representation of G:

(1) exp [- ~ Sp] = ~'. an~,(.gp)

At strong cupling the dominant term will be the one involving the character X0 of the trivial representation and without loss of generality we may normalize the action so that a0 = 1. After expanding the total Gibbs factor (which is a product over the plaquettes of our lattice) into a sum of terms having a definite representation at each plaquette, we group all plaquettes carrying nontrivial representations into connected sets called polymers. Integrating over the group produces a certain weight factor for these polymers and also leads to certain constraints for the representations occurring in adjoining plaquettes. A simple energy-entropy argument shows that for strong coupling these polymers are typically small and dilute, which allows to show convergence of the well-known strong-coupling expansion. As the coupling g decreases, the entropy of large polymers will eventually overwhelm the energy barrier for large polymers, leading to polymerization and triggering a bulk phase transition. While we have no rigorous proof that this polymerization actually occurs and, if so, triggers a phase transition, we think it is eminently plausible and supported by its phenomenological success, as we shall illustrate below (for more examples see also (4)). A plaquette carrying representation n will have a weight factor an. There will be a leading nontrivial represen~:ation, normally the one having the largest weight factor a~ < 1; the first polymers having a chance of becoming large will be simple closed surfaces built out of plaquettes carrying that representation. Such a surface of area A will have a weight (exponential of minus ,,energy~,) a~.

There will be of the order of c A such surfaces; we call A In c the entropy. As an example for which numerical data are available (~), let us discuss SU(2)

lattice gauge theory with a mixed action

1 1 (2) Sp = ~ ~,zf ~ ~z~ ,


where f refers to the fundamental and a to the adjoint representations of SU(2). We call the corresponding dual Gibbs factors af and aa. For small ~f and/~a both af and as are small and the polymer system is a dilute gas of small polymers. If we increase ~f at fixed fl~, surfaces carrying the fundamental representation will polymerize at some point. To estimate the entropy we compare with the U(1) gauge model which has its transition when the dual Gibbs factor is about 0.45. This leads to an estimate of flf crit given by al = 0.45. If we start again in the strong-coupling regime and increase fl~ eventually the Gibbs factor as will reach a critical value and there will be polymerization of surfaces carrying the adjoint representation. Before jumping to the conclusion that the critical value of this Gibbs factor is also near 0.45, one should remember that the product of the adjoint representation with itself again contains the adjoint representation. For this reason, (,adjoint~, surfaces may branch which leads to an increased entropy. The critical Gibbs factor therefore is reduced; comparison with the data of(13) show a mild reduction to about 0.36. If both af and aa are near their critical values, it will be likely that large surfaces containing both fundamental and adjoint representations are formed; this again leads to an increase in the entropy and therefore to slightly earlier occurrence of polymerization. Finally, we can contemplate the behavior of the system at large ~ , where it behaves essentially like a Z(2) gauge model at inverse coupling fir. Moving from small to larger flf we will encounter a polymerization threshold of surfaces built of plaquettes carrying the fundamental representation of Z(2), or almost equivalently, of SU(2). The critical value of flf should become almost independent of fig for large fl~ and be close to the critical fl of the Z(2) model, i.e. 0.44. In fig. 1 we compare the phase diagram obtained by the numerical data of('3) with the one predicted by our polymerization argument, using the level lines af=0.45 and a~=0.36. Considering the crudeness of our approximation in which we considered only one polymer carrying only one representation we think the agreement is quite good. A major, unavoidable difference is that according to our picture the line AB crosses the line fig = 0 and extends to infinity, just like in the corresponding mixed action U(1) model(~4).

Next we would like to present the numerical results we obtained for the SU(2) gauge theory with Wilson's action (in the computation SU(2) was approximated by the 120 elements subgroup Y). The computations were done on a lattice of size 64 and involved up to 20000 sweeps on 5 parallel lattices, using the double updating Metropolis algorithm of(3) on a Cray XMP. Motivated by our picture of the presumed phase transition we mainly studied a certain disorder operator that is sensitive to the abundance of the polymers described above. Our local disorder variable D is defined as follows: we take the product of exp [-/~Z] over

(1~) G. BHANOT and M. CREUTZ: Phys. Rev. D, 24, 3212 (1981). (,4) n . G. EVERTZ, J. JERS,~K, W. NEUHAUS and P. M. ZERWAS: Nucl. Phys. B, 251, 279 (1985).


Pc~

4.0

3.0

2.0

1.0

l

l

\ \

%\ A

\

I 0 1.0

\ \

\ \

2.0"\ ~f 3.0

\s Fig. 1. - Phase diagram of the mixed action SU(2) lattice gauge theory; the solid lines are based on the numerical data(5), the dashed lines on our energy-entropy estimates.

the six plaquettes forming the boundary of a 3-cube. We measure the expectation value as well as the susceptibility of the truncated 2-point function of D. Finally we measured the (untruncated) susceptibility of the Polyakov loops. The measurements were repeated for the U(1) gauge theory, with the group approximated by Z(10). The results for (D>, Z~ and Ze are shown in fig. 2 and 3. For U(1) we observe the well-known deconfining transition near ~ = 1. This is in rough (10 percent) agreement with our energy-entropy argument, using the Z(2) gauge model to estimate the entropy. The transition manifests itself in a steep rise in Ze, as well as a sharp drop in <D> and ZD, indicating a first-order transition. For SU(2) the entergy-entropy argument suggests a transition near

= 2.1, using the U(1) model to estimate the entropy. The results shown in fig. 3 are qualitatively similar to the ones obtained for U(1); the drop in D and particularly zD is even more pronounced than for U(1), whereas the increase in zP is milder. The results indicate that the model undergoes a first-order deconfining transition around fl = 2.3 (we expect a mild variation of that value with the volume). [We should mention, however, that there are certain differences

234 A. PATRASCIOIU, E. SEILER~ V. LINKE and I. O. STAMATESCU

10 2

I i I

101

10 o 0.9 .0 1.1 fl

Fig. 2. - (D), Xo, ~P for the U(1) lattice gauge theory on a 6 4 lattice. The lines are drawn to guide the eye. �9 102D, x 10-1XD, O 101Zp.

between U(1) and SU(2) visible. For instance for U(1) at ~ = 1.01 there is a clear two-state signal in the action; this does not occur for SU(2). Obviously the action is not sensitive to the transition we are observing in SU(2).] If we consult the recent data by Engels et al. (9) on the finite-temperature phase transition, we observe that the location of the bulk transition we have detected coincides approximately with the location of the finite-temperature deconfining transition on a lattice of length N~ = 5 in the time direction. The standard belief is of course that the deconfining transition will move to ~ = ~ as the temperature T goes to zero, according to the asymptotic-freedom formula.

In fig. 4 we present data compiled from the works of several authors (1~12) for the location ~c of the ,,finite-temperature deconfinement transition, in SU(3) lattice gauge theory; we plot 1/N~ vs. g2, where NT is the size of the lattice in time direction and g2 = 6/~r Our energy-entropy estimate would suggest a bulk transition around fl = 6. Extrapolating the data up to N~ = 10 would give a zero- temperature value in general agreement with that prediction. The last points, corresponding to NT = 11 to 14, seem to indicate a change in slope that is generally taken to signify- the onset of asymptotic scaling. The crucial question, which we have addressed already in some previous papers (1~,16), is if these data, taken on

(1~) A. PATRASCIOIU, E. SEILER, I. O. STAMATESCU: Obtaining reliable numerical information close to a critical point, MPI-PAE/PTh 31/88. (16) A. PATRASCIOIU, E. SEIZER and I. O. STAMATESCU: Critical slowing down, Monte Carlo renormalization group and critical phenomena, University of Arizona preprint (1988).


102

10 i

10 0

10 -1

! I I I I I I I

I'-.

I I I I I I I I 2.0 2.2 2.4- fl 2.6

Fig. 3. - <D>, ZD, ZP for the SU(2) lattice gauge theory on a 64 lattice. The lines are drawn to guide the eye. �9 103D, x 10-2XD, �9 Xe-

0.50

0.25

0 .I0

I I

I I

I I

I I

r

I I

/

/ I I / l

0.9 ~2= 6/p

I I I I 0.7 1.1 1.3

Fig. 4. - Finite-temperature deconfining transition for the SU(3) lattice gauge theory (compiled from (6.11,12)). �9 ref. C~ x ref. C1), �9 ref. (12).

i6 - I l Nuovo Cimento B.


larger lattices and at larger beta, are not affected by critical slowing down. This is obviously an extremely important issue which will decide if the conventional picture or the scenario we are proposing is the correct one. Another possible way to disprove our conjecture would be to separate clearly the deconfining and the accepted bulk transitions in the SU(N) models for D >/4. According to the conventional picture these two transitions are totally unrelated, hence it should be possible to separate them even on smaller lattices, for instance by modifying the action. Thus it should be possible to observe the finite-temperature remnant of the bulk transition without experiencing deconfinement at the same coupling. In our picture the two transitions are both driven by the same polymerization mechanism, hence cannot be separated. As illustrated above, given the action, our picture predicts also the rough location of the transition. In (3) we used this idea to displace the location of an analogous transition in the 0(4) spin model in 2/9 freely. In particular, by completely forbidding jumps of the spin between neighboring sites by more than the minimal angle, we managed to make the weak-coupling phase extend all the way to/3 = 0. Similarly, if in a lattice gauge theory we choose an action that agrees with Wilson's for plaquette group elements closer to the identity than some given distance and is infinite otherwise, we predict that the model will be in the deconfined phase for all values of the coupling constant and the temperature.

�9 RIASSUNTO (*)

Si rieerea un meeeanismo generale reeentemente proposto per resistenza di transizioni di fase nel contesto delle teorie di gauge del reticolo. Esso porta alla previsione che tutte le teorie di gauge del reticolo a temperatura zero in D 1> 3 devono subire una transizione di fase; ~ il limite della transizione di deconfinamento a temperatura finita. Si presentano dati numerici che confermano la nostra affermazione per la teoria di gauge del reticolo SU(2) a quattro dimensioni.

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The phase structure of lattice Yang-Mills theories