On the 3-d 3-state Potts model

Nuclear Physics B (Proc . Suppl .) 17 (1990) 335-338 North-Holland

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THE 3-d 3-STATE POTTS

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Rajiv V. GAVAI' Theory Division, CERN, 1211 Genève 23, Switzerland and Theory Group, T.I.F.R., Homi Bhabha Road, Bombay 400005, India We present results from high statistics investigations on the 3-d 3-state Potts model with and without an i er romagnetic next-to-nearest neighbour (nnn) couplings. On L3 lattices, with L ranging from 20 to , find a finite size scaling behaviour in all global observables which is characteristic of a first order phase transition in both the cases . Our investigations of the correlation lengths suggest it to be difficult to discern about the er of the phase transition from them alone.

1.

INTRODUCTION A nice and positive aspect of last year's controversyl~2 about the order of the deconfinement phase transition in SU(3) gauge theory has been a re sultant thorough and careful investigation of the various parts of the universality argument which led to the prediction 3 of a first order deconfinement phase transition in this theory. A major assumption in this argument is that of the lack of Z(3) criticality. One can motivate it in several ways, e.g., by using renormalization group arguments or early numerical simulations of the 3-d 3-state Potts model. However, none of these are very convincing. Indeed, the numerical simulations were performed on small lattices with modest statistics and it has been claimed that an addition of an irrelevant coupling is sufficient to challenge the renormalization group arguments. In particular, a numerical simulation of the Potts model with antiferromagnetic next-to-nearest neighbour coupling yielded evidence for a second order phase transition 4,5 . Motivated by all this, we undertook high statistics investigations of Potts models with and without nnn-coupling. We employed finite size scaling theory to determine the order of the phase transition by studying various thermodynamic observables . Since the claim of a second order phase transition in SU(3) gauge theory and the Potts model with nnn antiferromagnetic coupling were based on the observation that respective correlation lengths seemed to diverge as the size of the system, we also measured the correlation functions in our simulations and extracted

correlation lengths from them for both the models. e find a first order phase transition in both the cases of Potts model studied. While our analyses of both the thermodynamics and correlation length point towards this conclusion, we find the former analysis much more definitive and conclusive. The organization of this paper is as follows . In e next section we present the formalism beneath our analysis, including various definitions . Section 3 contains our results and the final section summarizes our conclusions . For lack of space, e ill present only some important results here; the interested reader will find more results, which were presented in the conference and which support our conclusions presented here, in our longer papers6}7. 2. - THE MODELS The Hamiltonian for the Potts models with a relative strength of the antiferromagnetic nnn coupling y is given by

where °i(k) = 0, 1 or 2 define the spins on site j or k: sj = exp(27riaj) and the sums run over nearest neighbour (nn) and next-to-nearest neighbour (nnn) pain respectively. The corresponding partition function is given lay

'Based on work done in collaboration with F. Karsch and B. Petersson 0920-5632/90/$3.50 © Elsevier Science Publishers B.V. North-Holland

Z

= E e-ß8 (ri)

.

(2.2)

R. V. Gavai/ The 3-d 3-state Potts mode!

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0.6 0.5 0.4 0.3 02 0.1 0

F re 1: The order parameter for y = 0.0 as a function of ,8 on L9 lattices. 'e used the Metropolis algorithm to simulate these models on L3 lattices with L = 12, 20, 24, 30, 36 and 48 for y = 0.0 and 20, 24, 32, 40 and 48 for y = 0.2, here the latter case has been claimed to have a second order phase transition4,s. .Typically, we performed 100 iterations to al for thermalization and then measured physical observables after every 10 iterations over 0.5-5.0 million iterations. Here will focus on the following observables only: i)The average order parameter , where S is defined by 3 1 (2.3) S = Zmax(no, nt, n2) - Z with n. given by ram -- V E, öo®, for a = 0, 1, 2, ii) the corresponding susceptibility, X, defined by --2) , (2.4) ( and iii) the correlation functions r(r), defined by 1

r(r) = 6L <

L

$ii> > ,

(2.5)

here s; = L'a ~exp(27ria) is the average spin on the plane i and r = ji - il denotes the distance beeen planes along one of the principal axes of the lattice. There are various ways of extracting correlation lengths from the correlation functions. We follow here the method of obtaining distance-dependent

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Figure 2: Same as Figure 1 but for -y = 0.2 and for corresponding lattice sizes mentioned in the text. masses from the ratios of these correlation functions at successive distances for a given lattice size and coupling. eking an ansatz that they decay exponentially, one can solve the transcendental equations for these ratios numerically to obtain the distance-dependent masses. If the ansatz is a good description of the data then the distance-dependent masses reach a plateau for large r from which one can read off the inverse correlation length at the given P and L. 3.

RESULTS

Figures 1 and 2 display our results for the average order parameter on the lattices studied for y = 0.0 and 0.2 respectively. Both are very suggestive of a discontinuity in the thermodynamic limit, since the finite sire effects seem to maize the order parameter larger(smaller) on smaller lattices for ,B < P,(# > #,). It is also interesting to note that the onset of finite size scaling behaviour seems to occur much later for y = 0.2, since the critical region appears to shift irregularly for L < 32 and only for L > 32 does one see the anticipated shift of P, towards larger P, as V -r oo. The first order nature of these transitions is also indicated by the presence of a two-state signal in the critical region . A typical run for y = 0.2 and P = 1.1901 on the 433 lattice is shown in Figure 3. One sees there that

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T/2000 Figure 3: The time evolution of the order parameter S for P = 1.1901 and -y = 0.2 on a 483 lattice. the order parameter S jumps from an ordered state to a disordered one and vice versa rather abruptly. We have also seen the anticipated finite volume dependence of such runs in the critical region : The interval of 0 in which one observes this increases as V decreases and there are more nips too. Figure 4 shows our results iu< Une magnetic susceptibility, X, as a function of ß for y = 0.2. It is evident that for L > 32 the peaks scale as V. One also sees the irregular behaviour noted earlier in form of shifts of the locations of these peaks which are varying in sign up to L = 32. But for L > 32 one obtains the anticipated behaviour. Our studies of the energy density, the specific heat and a higher order cumulant of the energy density, further reinforce our conclusions reached hereV. In particular, the estimates of #,, obtained from these quantities are in good agreement with those obtained from X above and the cumulant has a dip which seems to survive the thermodynamic limit. Finally. we display in Figure 5 the inverse correlation lengths on various lattices as a function of 0 for y = 0.2 . The corresponding figure for -y = 0.0 is very similar, except again for an early onset of finite size scaling behaviour. One sees a crossing of the data points for any two lattice sizes in the critical region which is (3 consistent with the expectation of the phase transition being first order. However, it is also clear from this figure that apart from the above qualitative indication

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Figure 4: The magnetic susceptibility, X, as a function of P for "y = 0.2. of the order of the phase transition, it is difficult to establish the order quantitatively. The presence of a huge tunnelling correlation length for P > Pc ma s an unambiguous determination of the physical correlation length impossible in the interesting critical region . A firm conclusion about the behaviour of the physical correlation length as the size of the system increases is therefore not possible . One can attempt to obtain the corresponding critical index from the data in Figure 5. However, it is not clear even theoretically what is should be for a first order phase transition . Also due to the visible sharpness near ß, in Figure 5 the numerical determination of the critical index depends strongly on the range of couplings assumed to be in the scaling

region . One possible way to define the physical correlation length is to use the cohnected correlation function . Unfortunately, on finite lattices it cannot be defined unambiguously. Nevertheless, it has been suggested8 that one should split the data for correlation functions in the critical region corresponding to the two phases visible rPhys in Figure 3 and then average. One then defines as

r(r) Phys = r(r)- < S>2 ,

.1)

and assumes to be zero in the confined phase. The physical correlation length can now be obtained

R. V. Gavai/ The 3-d 3-state Potts model

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C

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® 48 1184

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ß F re 5: The in correlation len h as a function _ of ß for y 0.2 for the lattices sizes studied .

from l"Phys by using the

method described in Sec. 2. Figure 6 displays our results for y = 0.0 obtained in this manner. One sees at the physical correlation length remains unchanged at the critical points of the larger lattices, even though the lattice size grows by a factor of o. It has a discontinuity on our largest lattice which agrees with the results of ref.8 but one also sees indications in Figure 6 that the discontinuity may vanish in the thermodynamic limit. It is, therefore, still an open question whether the physical correlation length in the 3-d 3-state Potts model has a discontinuity even if one accepted the prescription of ref. 8 as the right one. 4.

CONCLUSIONS

Our high statistics investigations of 3-d 3-state Potts. model with and without antiferromagnetic rararac pling have yielded strong indications that the phase transition is of first order in both the cases, although a comparatively late onset of finite size scaling is seen in the later case . We find the study of global observables using finite size scaling theory much more conclusive and definite . There are also indications of the first order nature of the phase transition in the correlation length data, which is, however, difficult to analyse quantitatively, especially since there is no clear picture of what the corresponding critical index should be .

Figure 6: The physical correlation length, obtained from rPhy$ defined in eq .(3.1), as a function of 8 for -y = 0.0 for the 243 , 363 and 483 lattices . 5.

ACKNOWLEDGEMENTS It is a great pleasure to acknowledge here the stimulating discussions I had with my collaborators F. Karsch and B. Petersson which helped me a lot in shaping my ideas on this subject. y participation in the conferfacilitated ence was greatly by the financial grant from the organisers . I would like to thank them sincerely, especially Prof . E. Marinari, for this kind gesture. REFERENCES 1. P. Bacilieri et al., Phys. Rev. Lett . 61L (1988) 1545 . 2. F.R . Brown, N.H . Christ, Y. Deng, M. Gao and T.J. Woch, Phys. Rev. Lett . 61L (1988) 2058 . 3. L.G . Yage and B. Svetitsky, Phys . Rev. (1982)963. 4. F. Fucito and A. Vulpiani, Phys. Lett . 69 33 . 5. L.A . Ferndndez et al., Phys . Lett . 308.

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6. R.V . Gavai, F. Karsch and B . Petersson, Nucl . 322 (1989) 738. Phys . 7. R.V . Gavai and F. Karsch, CERN preprint CERNTH .5553/89, September 1989. 8. M. Fukugita and (1989)13.

. Okawa, Phys. Rev. Lett . 63