The q-state gauge Potts model Pq in d-dimensions has been studied using Monte Carlo techniques. For d=2 no phase transitions were detected. The P2 model for d=3 shows a second order phase transition. On the other hand, all the d=3 (q≠2) and d=4 cases studied show first order phase transitions. In these cases, it was possible to estimate transition coupling parameters as well as latent heat. For selected cases, a study of the behavior of the Wilson loop factor was done.Facultad de Ciencias Exacta

Fanchiotti, Huner

Sciutto, Sergio Juan

Vucetich, Héctor

The European Physical Journal B

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Z. Phys. B - Condensed Matter 55, 57-60 (1984) Condensed Zeitschrift Matter for Physik B 9 Springer-Verlag 1984 Monte Carlo Calculations on the Gauge Potts Model* H. Fanchiotti, S.J. Sciutto, and H. Vucetich Laboratorio de Fisica Te6rica, Departamento de Fisica, Universidad Nacional de La Plata, La Plata, Repfblica Argentina Received December 28, 1983 The q-state gauge Potts model Pq in d-dimensions has been studied using Monte Carlo techniques. For  d = 2 no phase transitions were detected. The P2 model for d--3 shows a second order phase transition. On the other hand, all the d = 3  (q=t=2) and d = 4  cases studied show first order phase transitions. In these cases, it was possible to estimate transition coupling parameters as well as latent heat. For selected cases, a study of the behavior of the Wilson loop factor was done. 1. Introduction In studying simple gauge models, one can acquire a great amount of experience that could be useful when attacking more realistic (and generally, more difficult) models. The gauge Potts model [1,2] is certainly a very good example of such a case. There exist several attempts, and different approaches, for solving this model, such as series expansion [1, 2, 3, 4] mean field techniques [5, 6], and variational methods [7]. However, it is worthwhile mentioning that different approaches give in same cases, dif- ferent results. The main purpose of this paper is to report the results of Monte Carlo [8] calculations on the q- state gauge Potts model, Pq, in d-dimensions, for q up to 12 and d=2 ,  3 and 4. The previous Monte Carlo studies of the Potts mod- el, refer to the case of a global symmetry [9]. There are also studies of more general models which re- duce to a Potts model in particular cases [10]. We present a more exhaustive study of the model, which includes the analysis of the q-dependence of tran- sition coupling parameters and latent heats, and the influence of the dimensionality of the lattice on the phase structure. Moreover, we have done a study of the behavior of the corresponding Wilson loop [11] in several cases. In Sect. 2 there is a brief description of the model, together with the definitions of the * Partially supported by CONICET, CIC Pcia. de Buenos Aires and SUBCYT, Argentina magnitudes used throughout the paper. Section 3 contains the results of the calculations and our con- clusions are included in Sect. 4. 2. The Model The euclidean action of the gauge Potts model is S = - / 3  ~ 6v~.1 (1) plaq. where plaq. denotes a primitive square (plaquette) on the d-dimensional lattice, /3, is the coupling pa- rameter, Up= U 1 U 2 U 3 U 4 is the product of link vari- ables U (taking the values exp(i.2rc.k/q), k=0,  1, 2, ..., q - 1 )  around the plaquette and 6 is the standard Kronecker symbol. The action (1) allows the definition of the partition function: Z(fi)= ~, exp(-S(conf.))  (2) {conf.} where the sum runs over all possible configurations of the system, that is, all possible values of the link variables U i. The magnitudes of interest of Monte Carlo calcu- lations are: The average action per plaquette (the "internal en- ergy") that reads: 58 H. Fanchiotti et al.: Monte Carlo Calculations on the Gauge Potts Model <s> E - ( 3 )  Np with N p  being the number of plaquettes on the lat- tice. The "specific heat" C=-fl2 ~ (fl)=(<S2)-(S)2)/Np (4) And the Wilson loop W, defined by W=<Re(U 1U 2 ... U,)> (5) where the links 1, 2, ..., n form a closed loop in the lattice. We recall that at a first order phase transition, the internal energy E, is discontinuous in /3. On the other hand, at a second order one E is continuous and the specific heat C, diverges if the lattice is infinite. For finite lattices, C presents a peaked curve which sharpens as lattice size increases. It is also known that in an infinite lattice the Wilson loop W approaches zero when the loop is increased, follow- ing an area (perimeter) law in a disordered (ordered) phase [11, 12]. This behavior can be observed in a finite lattice if the size of the loops considered is small when compared with the total lattice size. 3. Results We present our results for the Potts model for the already mentioned values of q, and for two, three or four-dimensional lattices with periodic boundary conditions. All the Monte Carlo computations were done using the Metropolis algorithm [13]. First we have determined the phase structure of the different systems. The "thermal cycle method" [12] (calculat- ing E for different values of fl as usual) was used to detect first-order phase transitions, and the "specific heat divergence" method [14] for second order ones. Neither first nor second order phase transitions were detected in the two dimensional systems, in agree- ment with previous theoretical results for infinite lattices [15]. We should mention that q=2,  3, 4, 5, 6, 8, 10, 12; N2=362 and f l~[0 ,5]  were the values of q, the lattice size and the interval of/3 considered, respectively. The P2 model, being equivalent to the Z 2 model has been widely studied [12, 14, 16] and our results are in agreement with these previous analysis. For  ex- ample, in d=3 ,  P2 shows a second order phase tran- sition for/3c = 1.400 _+ 0.025. All the other cases studied in three and four dimen- sions show first-order phase transitions. The results Table 1. Geometric parameters, fl-interval and transition parame- ters for the three-dimensional systems studied q N fl-interval tic-interval AE l 3 11 [0.01; 3.00] [1.43; 1.90] 4 11 [0.01; 3.00] [1.43; 1.97] 6 11 [0.01; 4.00] [1.59; 2.15] 8 1i [0.01; 5.003 [1.63; 2.38] 10 11 [0.01; 5.00] [1.70; 2.38] 12 11 [0.01; 5.00] [1.73; 2.51] 0.216 +-0.037 0.318 +-0.047 0.49 _+0.10 0.65 +-0.13 0.76 +_0.10 0.95 +0.14 Table 2. The same, but for four-dimensional systems q N /Mnterval /~c-interval AE l 3 8 [0.01; 3.00] [0.83; 1 .33]  0.315+__0.047 4 6 [0.01; 3.00] [0.87; 1 .59]  0.429+__0.067 5 8 [0.01; 4.00] [0.92; 1 .70]  0.517__+0.079 6 6 [0.01; 4.00] [0.98; 1 .74]  0.610+_0.081 8 6 [0.01; 5.00] [1.08; 1 .89]  0.740+_0.084 10 6 [0.01; 5.00] [1.07; 2.13] 0.88 __+0.10 12 6 [0.01; 5.00] [1.17; 2.20] 0.90 __+0.10 are summarized in Tables 1 and 2. The transition values for the coupling parameter and the corre- sponding latent heat, AE z were obtained from direct measurement on the thermal cycles, and for that reason they should be regarded as rough esti- mations. In d = 4  the gauge Potts model is self-dual allowing the parametrization of the coupling parameter in the self-dual point by [3]: fl~ = ln(1 + @) (6) Figure 1 shows both the results obtained from our Monte Carlo calculations and the self-dual predic- tion (6). There we have also included an accurate estimation for q = 6. This estimation was obtained in the following way: First, long runs with mixed initial configurations [16] were used to evaluate E, C and W, for values of/~ belonging to the "transition in- terval" obtained with the thermal cycle method, la- belled /3~-interval in Tables 1 and 2. Then, accurate bounds for ]~ were obtained by taking into account the discontinuity of E and the change in the be- havior of W. The results obtained for the latent heat together with the values coming fi'om the 1/q-expansion [2]: AE~=I 20/9 1412/675 1.239091001 - ~ + . . .  ( 7 )  q q2 q3 are shown in Fig. 2. H. Fanchiotti et al.: Monte Carlo Calculations on the Gauge Potts Model 2~0- 1,5- 1.2- 0,8- 0,4- 0 , I ~ I E ll2 2 4 8 10 ~ Fig. 1. Transition coupling parameter vs. q for four-dimensional systems. The vertical bars represent the thermal cycle bounds and the solid curve is the self dual prediction (6). The square dot at q = 6 shows an accurate estimation for fl, 59 e ' '  ),E O,l . e. " ' // 9 ' /1 3 , ;  " 9 9 / / /  9 " e  / /  9 " / / m  e.  " ~ r  ~ 0,6 1,2 1,8 2,4 /# Fig. 3. Wilson loop factor vs. /~ for the P4 model (d = 2). Circular and square dots show the values of W for single and double plaquette, respectively. The solid (dashed) line represents the pe- rimeter (area) law prediction AE[ 1,0" O , S -  i 0,6- 0,4- 0,2 L+ I I I Fig. 2. Latent heat vs. q for four-dimensional systems. The circles show the values obtained from thermal cycle calculations and the solid curve is the 1/q-expansion (7) Lack  of c o m p u t a t i o n a l  t ime m a d e  imposs ib le  the ca lcu la t ion  of  a comple te  set of  accura te  es t imat ions  for/~c and  A E  I. Final ly ,  we have s tudied in detai l  the behav io r  of the Wi l son  loop  for the mode l  in two and th ree-d imen-  s ional  lat t ices wi th  36 and  9 sites per  d imens ion  respectively,  and  var ious  values of  q. F igures  3 and 4 show the results  ob t a ined  for q = 4 .  The  lines iden-  tified as "pe r ime te r  law" and " a r e a  l aw"  were ob-  ta ined  by p lo t t ing  the 2nd and  the 4th power  of the cor respond ing  single p laque t t e  value,  respectively9 The  square  poin ts  show the values ob ta ined  for a doub le  plaquet te ,  that  is a square  " p l a q u e t t e "  with two links at  each side. The  da t a  ob ta ined  for d = 2  (Fig. 3) show an a rea  law behavior .  In  the d = 3 case (Fig. 4), the existence of  a f i rs t -order  phase  t rans i t ion  },6 1,4 ,* 9 'e'"" / ' [ .'e" 0,2 9 , - ~  I . '" /2/11 9" / 0,5 1,0 1.5 Fig. 4. As Fig. 3, but for d = 3 I 2.0 shows up in the change in the behav io r  of  W. Fur -  thermore ,  it can be easily seen the agreement  with an a rea  law in the d i so rde red  phase  ( lower values of /~) and  with a per imeter  law in the o rdered  one. The  t rans i t ion  be tween one law to the o ther  takes  place in a small  n e i g h b o u r h o o d  of the cri t ical  pa r ame te r  ]?c. 4. Conclusions W e  have carr ied out  an exhaust ive s tudy of the phase  s t ructure  of  the gauge Pot t s  mode l  by means  60 H. Fanchiotti et al.: Monte Carlo Calculations on the Gauge Potts Model of M o n t e  Car lo  computa t ions .  The  results ob ta ined  are useful to check a p p r o x i m a t e  methods .  If qc is the cri t ical  q -pa ramete r  (that  is, the pa r t i cu la r  value of q with the p rope r ty  tha t  for any  given q>qc, the Pq mode l  has a f i rs t -order  phase  transi t ion),  we conf i rmed that  qc < 3 for d = 3 and  qc < 2 for d = 4, in agreement  with 1/q-expansion pred ic t ions  [2, 4]. I t  is i m p o r t a n t  to r e m a r k  that  the mean  field a p p r o x i m a -  t ion [6] predic ts  a f i rs t -order  phase  t rans i t ion  for the P2 mode l  with d = 3 ,  showing the l imi ta t ions  of that  method.  However ,  the mean  field results  for the la tent  hea t  are in good  agreement  wi th  our  data.  Studies abou t  the influence of  finite la t t ice size and  b o u n d a r y  condi t ions  as well as the inclusion of bo-  sonic and fermionic  mat te r  fields in the act ion,  should  be deve loped  in future works.  We are indebted to C.A. Garcia Canal for critically reading the manuscript, and to L.N. Epele and C. Camarata for enlightening discussions. One of the authors, S.S., wants to thank R. Gianotti for useful conversations about Monte Carlo techniques. References 1. Kogut, J.B.: Phys. Rev. D21, 2316 (1980) 2. Kogut, J.B., Pearson, R.B., Shigenitsu, J., Sinclair, D.K. : Phys. Rev. D22, 2447 (1980) 3. Ginsparg, P., Goldschmidt, Y.Y., Zuber, J.B.: Nucl. Phys. B170 [FS1], 409 (1980) 4. Kogut, J.B., Sinclair, D.K.: Solid State Commun. 41, 187 (1982) 5. Katec@, R.: Commun. Math. Phys. 82, 391 (1981) 6. Camarata, C., Epele, L.N., Fanchiotti, H., Garcia Canal, C.A.: Universidad Nacional de La Plata preprint (1983) 7. Masperi, L., Omero, C.: Nucl. Phys. B200 [FS4], 121 (1982) 8. Binder, K.: In: Phase transitions and critical phenomena. Domb, C., Green, M.S. (eds.), Vol. 5B. New York: Academic Press 1976; Creutz, M., Jacobs, L., Rebbi, C.: Phys. Rep. 95, N ~ 4 (1983) 9. B16te, I-I.W.J., Swendsen, R.H.: Phys. Rev. Lett. 43, 799 (1979) 10. Fukugita, M., Kobayashi, M., Kaneko, T.: Nucl. Phys. B215 [FS7], 289 (1983) 11. Wilson, K.G.: Phys. Rev.D10, 2445 (1974) 12. Creutz, M., Jacobs, L., Rebbi, C.: Phys. Rev. D20, 1915 (1979) 13. Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E.: J. Chem. Phys. 21, 1087 (1953) 14. Jongeward, G., Stack, J., Jayoprokash, C.: Phys. Rev. D21, 3360 (1980) 15. Kadanoff, L.P.: Rev. Mod. Phys. 49, 267 (1977) 16. Creutz, M., Jacobs, L., Rebbi, C.: Phys. Rev. Lett. 42, 1310 (1979) H. Fanchiotti S.J. Sciutto H. Vucetich Departamento de Fisica Laboratorio de Fisica Tedrica Universidad Nacional de La Plata C.C. 67 1900 La Plata Repfiblica Argentina 

Monte Carlo calculations on the gauge Potts model

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Monte Carlo calculations on the gauge Potts model

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