The critical behaviour of the long-range Potts chain from the largest cluster probability distribution

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1 Physica A 314 (2002) The critical behaviour of the long-range Potts chain from the largest cluster probability distribution Katarina Uzelac a;, Zvonko Glumac b a Institute of Physics, P.O.B. 304, Bijenicka 46, HR Zagreb, Croatia b Faculty of Electrical Engineering, Kneza Trpimira 2B, Osijek, Croatia Abstract We present the numerical study of the one-dimensional Potts model with power-law decaying ferromagnetic interactions. The largest cluster probability distribution is obtained by Monte Carlo simulations using the Swendsen Wang cluster algorithm with cumulative probabilities. The nite-size scaling analysis of the largest cluster is used to derive the critical behaviour in the non-classical regime of this model for various values of q. c 2002 Elsevier Science B.V. All rights reserved. PACS: q; Cn Keywords: Phase transitions; q-state Potts model; Long-range interactions; Monte-Carlo simulations The models involving long-range interactions have an important role in describing many complex systems, from physics to economy or biology, but the equilibrium critical phenomena in these models are still not well understood and deserve further attention. We consider here the 1d Potts model with long-range interactions dened by the Hamiltonian H = i j J i j 1+ s i;s j ; (1) Corresponding author. Tel.: ; fax: address: katarina@vrabac.ifs.hr (K. Uzelac) /02/$ - see front matter c 2002 Elsevier Science B.V. All rights reserved. PII: S (02)

2 K. Uzelac, Z. Glumac / Physica A 314 (2002) where J 0, s i is a q-state Potts spin at the site i, 0, is the Kronecker symbol and the summation is over all the pairs of the system. Further in text we use J =k B =1. Even in one dimension, this paradigmatic model involves variety of critical regimes [1 4] in rather complicated (q; ) diagram, which contains most features encountered in (q; d) plane of the same model with short-range interactions [5]. Apart from the precise determination of the critical behaviour in the non-trivial regime, number of questions are still open, as for instance the position of the borderline between the rst- and second-order phase transitions [4,6 8]. Due to non-locality of interactions and its discrete symmetries, model is not easily accessible by conventional renormalisation-group or Monte Carlo (MC) approaches, and more dedicated approaches [9,1,10,2], were developed and are used in complementary way. We present here an approach based on cluster statistics obtained by numerical simulations. In standard MC procedures the long-range interactions are dicult to handle since the number of operations per spin ip is proportional to the size of the system. To avoid this we use the cumulative probabilities, recently proposed by Luijten and Blote [10], which reduces the computing time by several orders of magnitude and opens the possibilities to more thorough numerical studies of the long-range models. We implemented the cumulative probabilities to the Swendsen Wang cluster algorithm [11] to explore more extensively the cluster statistics, which is mostly used in geometrical transitions such as percolation, but may be applied to the thermal transitions as well [12]. We limit in the present account to results for the non-trivial critical behaviour (including = 1) derived from of the largest cluster probability distribution. It is dened as P max (l)= 1 N max (l) ; (2) n MCS where N max (l) is the total number of occurrences of the largest cluster of size l during n MCS MC steps of the system. The corresponding rst and second moments are respectively related to the order parameter and susceptibility. The Binders cumulant ratio [13], dened in the same way as for the standard order parameter R L (T)= l4 l 2 2 ; (3) is used to derive simultaneously the critical temperature and exponent. Since the the ratio R L (T) att c does not depend on L, T c can be identied as a common crossing of curves R L (T) for dierent L (see Fig. 1). The error bars are less than 0.2%. The critical exponent is derived by overlapping the curves, using the scaling property R L (T)=f(L 1= ), where =(T T c )=T c (see Fig. 2). The results for q = 2 and 3 are summarised in Tables 1 3. The case q = 2 which corresponds to Ising model, may serve as a test, since the results by several alternative approaches [14 16,1,17] exist, although with varying precision. The results for T c, which are obtained with precision better than 0.02% agree well with the results of nite range scaling (FRS) [1] and coherent anomaly [18], the most precise of the cited results. The existing results for the critical exponent dier even more, so that dierences among them largely exceed the limits of precision of our results estimated

3 450 K. Uzelac, Z. Glumac / Physica A 314 (2002) R L (T) T/J Fig. 1. R L (T ) versus (T T c) for (q =3, =0:9) for sizes L from 1000 to R L (T) τ N 0.48 Fig. 2. Plots of R L (T ) from Fig. 1, but versus scaled temperature. Table 1 The largest cluster results (LC) for T c for q = 2 compared to earlier results LC [14] [16] [1] [17] [18] 0.6 1:770 ± 0: :463 ± 0: :2155 ± 0: :001 ± 0:

4 K. Uzelac, Z. Glumac / Physica A 314 (2002) Table 2 The largest cluster results (LC) for for q = 2 compared to earlier results LC [14] [15] [16] [1] [17] 0.6 0:50 ± 0: :51 ± 0: :47 ± 0: :40 ± 0: Table 3 The largest cluster results (LC) for T c and for q = 3 compared to earlier results T c(lc) T c [1] T c [17] T c [18] (LC) [1] 0.8 1:026 ± 0: :61 ± 0: :8735 ± 0: :48 ± 0: up to 4%. Agreement is best with results obtained by FRS [1]. For a more detailed comparison, see Ref. [8]. For q = 3, the results permitted us to check the intriguing suggestion by Cannas et al. [17], that would not depend on q, and to conclude the contrary, in agreement with earlier FRS results. A dierent procedure was applied in the case = 1, characterised by the Kosterlitz Thouless type of transition for all q [3]. The correlation length there exhibits the essential singularity of the form exp[const:=(t T c ) p ], where p=2=(q+2) [3]. Similar holds for specic heat and susceptibility. The starting point is to dene a function as ; (4) N =const: by keeping constant the normalised susceptibility N instead of a normalised gap. The normalised susceptibility is given by N (T )= N (T)=N, where the susceptibility N (T) is obtained as a second moment of the largest cluster distribution while we use the exact expression =2 for the critical exponent in evaluating the anomalous dimension of the susceptibility. Then, following Roomany and Wyld [19] and discretizing the above derivative, we obtain the expression 2ln[ (T)= N (T )= M (T )] ln(n=m)[ N (T )= N (T )+ M (T )= M (T)] ; (5) for the -function of the innite system. For the critical behaviour in the form of essential singularity cited above, (T ) (T T c ) 1+p. In Figs. 3 and 4 are presented the preliminary results for q = 3and5 ( = 1), obtained for modest sizes of sites as the log-log plots of versus (T T c ). The resulting exponent p is equal to 0:36 ± 0:04 and 0:24 ± 0:04 for the cases q =3 and 5, respectively, which is very close to the values of p obtained for the correlation length by Cardy [3].

5 452 K. Uzelac, Z. Glumac / Physica A 314 (2002) L,M = 1000,2000 L,M = 2000,3000 lin. fit of 1000,2000 lin. fit of 2000,3000 ln(β) ln[(t T c )/J] Fig. 3. The log log plots of -function versus (T T c) for q =3, = 1 and T c =0:71. 2 L,M = 1000,2000 L,M = 2000,3000 lin. fit of 1000,2000 lin. fit of 2000, ln(β) ln[(t T c )/J] Fig. 4. The log log plots of -function versus (T T c) for q =5, = 1 and T c =0:64. In conclusion, we have shown on few examples that the numerical study of cluster statistics can provide a useful insight into rather complicated critical behaviour of the Potts model with long-range interactions providing accurate results for critical temperature and exponents. The study includes the procedure to deal with the case =1 characterised by the essential singularity. This approach allows wider study of cluster

6 K. Uzelac, Z. Glumac / Physica A 314 (2002) structure and statistics which in future may shed some light to the still open problems such as the borderline between the rst- and second-order phase transition. References [1] Z. Glumac, K. Uzelac, J. Phys. A 26 (1993) [2] E. Luijten, H.W. J. Blote, Phys. Rev. B 56 (1997) [3] J. Cardy, J. Phys. A 14 (1981) [4] Z. Glumac, K. Uzelac, Phys. Rev. E 58 (1998) [5] F.Y. Wu, Rev. Mod. Phys. 54 (1982) 235. [6] M. Kretch, E. Luijten, Phys. Rev. E 61 (2000) [7] E. Bayong, H.T. Diep, V. Dotsenko, Phys. Rev. Lett. 83 (1999) 14. [8] K. Uzelac, Z. Glumac, Phys. Rev. Lett. 85 (2000) [9] Z. Glumac, K. Uzelac, J. Phys. A 22 (1989) [10] E. Luijten, H.W.J. Blote, Int. J. Mod. Phys. C 6 (1995) 359. [11] R.H. Swendsen, J.-S. Wang, Phys. Rev. Lett. 58 (1987) 86. [12] A. Coniglio, W. Klein, J. Phys. A 13 (1980) 2775; M. D Onorio Meo, D.W. Heermann, K. Binder, J. Stat. Phys. 60 (1990) 585. [13] K. Binder, Phys. Rev. Lett. 47 (1981) 693. [14] J.F. Nagle, J.C. Bonner, J. Phys. C 3 (1970) 359. [15] M.E. Fisher, S.K. Ma, B.G. Nickel, Phys. Rev. Lett. 29 (1972) 917. [16] J.L. Monroe, R. Lucente, J.P. Hourlland, J. Phys. A 23 (1990) [17] S.A. Cannas, A.C.N. de Magalhães, J. Phys. A 30 (1997) [18] J.L. Monroe, J. Phys. A 32 (1999) [19] H.H. Roomany, H.W. Wyld, Phys. Rev. D 21 (1980) 3341.