Phase transition in the majority-vote model on the Archimedean lattices
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec Phase transition in the majority-vote model on the Archimedean lattices
Unjong Yu
Department of Physics and Photon Science, Gwangju Institute of Science and Technology, Gwangju 61005, South Korea ∗ (Dated: October 25, 2018)The majority-vote model with noise was studied on the eleven Archimedean lattices by the Monte-Carlo method and the finite-size scaling. The critical noises and the critical exponents were obtainedwith unprecedented precision. Contrary to some previous reports, we confirmed that the majority-vote model on the Archimedean lattices belongs to the two-dimensional Ising universality class. Itwas shown that very precise determination of the critical noise is required to obtain proper valuesof the critical exponents. PACS numbers: 05.20.-y, 05.70.Ln, 64.60.Cn, 05.50.+q
I. INTRODUCTION
The Ising model [1] has played a crucial role in the de-velopment of important concepts in the statistical physicssuch as the phase transition, critical phenomena, and theuniversality class [2, 3]. Although it was proposed to ex-plain ferromagnetism, it can be applied to various phe-nomena of material systems: liquid-gas systems at thecritical point, order-disorder transitions in binary alloysystems, charge-ordering in mixed-valence compounds,etc. In addition, recently, many kinds of varieties of theIsing model were proposed to explain social phenomena[4, 5]. For example, spin state of a site in the Ising modelmay represent a social opinion or preference of a person,which can be affected by his or her acquaintances. AnIsing-like model is also used to simulate racial segrega-tion, where different spin states represent people of dif-ferent races [6]. In this case, the total number of peopleof a race is constant and only the spacial configurationof people can change.Compared with material systems, the social applica-tion of the Ising model should take into account twopoints. The first one is that the structure of social in-teractions are usually complex networks rather than pe-riodic lattices [7]. It changes the character of the phasetransition of the Ising model [8, 9]. The second point isthat most of social phenomena are irreversible and outof equilibrium. Therefore, it is not trivial to apply theequilibrium statistical mechanics to the nonequilibriumsocial dynamics. Especially, it is not clear whether theuniversality hypothesis, which insists that systems of thesame spacial dimension and the symmetry of the orderparameter share the same critical exponents [10], is ap-plied also to nonequilibrium models.Interestingly, it was proposed that the stochasticnonequilibrium systems with up-down symmetry belongto the equilibrium Ising universality class in a steadystate [11]. If it is true, an Ising-like nonequilibriummodel in any kind of two-dimensional (2D) lattice shouldhave the same critical exponents as the equilibrium Ising ∗ E-mail me at: [email protected] model in the 2D square lattice [12]. It was confirmed ina few systems [13–15]. As for the majority-vote model(MVM), which is one of the well-studied nonequilibriummodels in the opinion dynamics with up-down symme-try, it was confirmed in the 2D square lattice [16, 17].However, later a few other works on other 2D lattices(triangular, honeycomb, kagom´e, maple-leaf, and bouncelattices) reported that critical exponents of the MVM aredifferent from those of the Ising model [18, 19]. More re-cently, it was argued again that the MVM on the triangu-lar and honeycomb lattices belongs to the Ising univer-sality class [20]. In the three-dimensional simple-cubiclattice, there is also controversy about the universalityclass: Ref. [21] reported different critical exponents of theMVM from the Ising universality class, but Ref. [22] ar-gued against that. Therefore, it is important to concludeabout the nature of the phase transition of the MVM onthe regular lattice.In this work, phase transitions and critical phenom-ena of the MVM are studied on the eleven Archimedeanlattices, which are 2D lattices by uniform tiling of regu-lar polygons. It was proved that there exist only elevenArchimedean lattices [23]: They are listed in Fig. 1 andTable I. Due to simplicity and various topologies, they aregood bases for systematic study on 2D systems [24, 25].The MVM on six Archimedean lattices has been stud-ied [16–20], and the results on the other five latticesare reported for the first time in this paper. By exten-sive Monte-Carlo calculations, the critical noise of eachArchimedean lattice was obtained with unprecedentedprecision, and it is confirmed that the critical exponentsof the MVM on all of the eleven Archimedean lattices aresame as the 2D Ising universality class. Possible reasonsfor the discrepancies of previous works are also discussed.
II. MODEL AND METHODS
The MVM used in this work is defined by the followingspin flip probability [16]: w ( σ i → − σ i ) = 12 − (1 − q ) σ i sgn X j =NN( i ) σ j , (1) TABLE I. Name, number of lattice points per basis ( B ), coordination number ( z ), number of next-nearest neighbors ( z ),critical noise ( q c ), critical exponents ( ν , γ , and β ), and references for the eleven Archimedean lattices within the majority-vote model. A next-nearest neighbor is a site with shortest-path-length of two from a given site. SHD and CaVO meansquare-hexagonal-dodecagonal and CaV O , respectively. Results of this work are in bold.Name B z z q c /ν γ/ν β/ν Ref.T1 Triangular (3 ) 1 6 12 0.114(5) 1.08(6) 1.59(5), 1.64(1) 0.12(4) [19]0.1091(1) 1.01(2) 1.759(7) 0.123(2) [20] T2 Square (4 ) 1 4 8 0.075(1) 0.99(5) 1.70(8) 0.125(5) [16]0.075(1) 0.98(3) 1.78(5) 0.120(5) [17] T3 Honeycomb (6 ) 2 3 6 0.089(5) 0.87(5) 1.64(5), 1.66(8) 0.15(5) [19]0.0639(1) 1.01(2) 1.755(8) 0.123(2) [20] T4 Maple leaf (3 ,
6) 6 5 9 0.134(3) 0.98(10) 1.632(35) 0.114(3) [18]
T5 Trellis (3 , ) 2 5 10 T6 Shastry-Sutherland (3 , , ,
4) 4 5 11
T7 Bounce (3 , , ,
4) 6 4 8 0.091(2) 0.872(85) 1.596(54) 0.103(6) [18]
T8 Kagom´e (3 , , ,
6) 3 4 8 0.078(2) 0.86(6) 1.64(3), 1.62(5) 0.14(3) [19]
T9 Star (3 , ) 6 3 4 T10 SHD (4 , ,
12) 12 3 5
T11 CaVO (4 , ) 4 3 5
2D Ising model (exact) 1 1.75 0.125 [12]
T1 T2 T3T4 T5 T6T7 T8 T9T10 T11
FIG. 1. The eleven Archimedean lattices. where the spin σ i at site i can have only ± i . Thefunction sgn( x ) is the sign function, which gives +1, − x , respectively.This update rule is a sort of death-birth dynamics, whereone site is chosen to forget its spin state and its nearestneighbors determine a new spin state of the site [26]. Inthe MVM, it follows the spin of the majority with prob-ability (1 − q ) and that of the minority with probability q . If the two kinds of neighbors tie, the spin is deter-mined at random. The parameter q is called the noise.For small q , one of the two spin states will prevail thesystem, and the system will fluctuate randomly withoutorder for q close to half. It was shown that there existsa continuous phase transition at the critical noise q c be-tween the two phases [16]. When q is larger than half,the site prefers to choose the minority spin of nearestneighbors and another phase transition can exist [27].In the case of equilibrium spin models, any Monte-Carlo dynamics that satisfies the detailed balance andthe ergodicity gives equivalent results if the simulationis properly performed. However, nonequilibrium mod-els depend on details of the update rule, and so cluster- update algorithms [28], which mitigate the critical slow-ing down, cannot be used. In addition, since there is noconcept of energy and Boltzmann distribution [17], ex-tended ensemble methods [29] also fail. Therefore, weperformed the simulation directly using the update ruleof Eq. (1). Most of the results were obtained by decreas-ing the noise q very slowly from q = 1 / × warming-up Monte-Carlo steps were followed by 10 steps for measurementnear the critical noise. Each site has a chance to flipits spin one time on average per one Monte-Carlo step.All the calculations were performed on parallelograms ofsize N = B × L × L , with the number of sites per ba-sis B and the number of bases in one direction L . Theperiodic boundary condition was used.At each temperature, the magnetization m , the mag-netic susceptibility χ , and the fourth-order Binder cumu-lant U are calculated as follows [16]. m = 1 N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X i =1 σ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (2) χ = N (cid:0) h m i − h m i (cid:1) , (3) U = 1 − (cid:10) m (cid:11) h m i . (4)In the definition of the susceptibility χ , sometimes a func-tion of q is multiplied [17, 20–22, 27], but critical expo-nents are not affected by that because the functions donot show critical behavior at q c . Since energy and spe-cific heat are not defined, only three critical exponents( β , γ , and ν ) are defined [30, 31]: m ( L, q ) = L − β/ν ˜ m [ L /ν ( q − q c )] , (5) χ ( L, q ) = L γ/ν ˜ χ [ L /ν ( q − q c )] , (6) U ( L, q ) = ˜ U [ L /ν ( q − q c )] , (7) dU ( L, q ) dq = L /ν ˜ U ′ [ L /ν ( q − q c )] , (8)where L = √ N is the linear size of the cluster and ˜ m ( x ),˜ χ ( x ), and ˜ U ( x ) are scaling functions. The critical noise q c is determined first, and the critical exponents are ob-tained from the physical quantities ( m , χ , and dU/dq )calculated at q c [16–22]. III. RESULTS AND DISCUSSION
The critical noise q c can be determined by the max-imum of the magnetic susceptibility χ or the derivativeof magnetization dm/dq . The peak position q c ( L ) de-pends on the linear cluster size L by q c ( L ) = q c ( L = ∞ ) + λL − /ν , where the parameter λ depends on the lat-tice type and the physical quantity measured [31]. How-ever, it is very difficult to locate the peak position pre-cisely without extended ensemble methods [29]. Alterna-tively, the critical noise can be found by the crossing of U q q L = 10 L = 14 L = 20 L = 30 L = 40 L = 60 L = 80 L =120 L =160 L =200 L =260 FIG. 2. (Color online) Binder cumulant ( U ) as a func-tion of noise ( q ) for the majority-vote model on the elevenArchimedean lattices with various linear sizes ( L ). The criti-cal noises ( q c ) obtained by the crossing of the Binder cumulantare shown in vertical dashed lines. the fourth-order Binder cumulant, because the Binder cu-mulant becomes independent of the cluster size at q = q c by Eq. (7), ignoring the correction critical exponent ( ω )[32]. The error from the ignorance of the correction crit-ical exponent is less than statistical error in this calcu-lation. Figure 2 shows the Binder cumulant close to thecritical point, and the critical noises obtained by thismethod are listed in Table I. The magnetic susceptibilitydata in Fig. 3 support the results: The maximum suscep-tibility position approaches q c with increasing the clustersize and it is very close to q c for large clusters. As is ex-pected, a lattice with more nearest neighbors tends tohave higher critical noise, but differently from the Isingmodel, there are some exceptions [20]. When the numberof nearest neighbor is same, a lattice with more numberof next-nearest neighbors has larger critical noise. (Here,a next-nearest neighbor means a site with shortest-path-length of two from a given site.) c q q L = 10 L = 14 L = 20 L = 30 L = 40 L = 60 L = 80 L =120 L =160 L =200 L =260 FIG. 3. (Color online) Magnetic susceptibility ( χ ), which isdefined in Eq. (3), as a function of noise ( q ) for the majority-vote model on the eleven Archimedean lattices with variouslinear sizes ( L ). The critical noises ( q c ) obtained in Fig. 2 arerepresented by vertical solid lines. Our results of the critical noise are consistent withRef. [16, 17, 20], but those of Ref. [18, 19] are much largerthan ours out of error bars. They used smaller numberof Monte-Carlo steps (2 × ) for warming-up at eachtemperature, but we confirmed it is enough for moderatesize clusters ( N .
2 3 4 5 6 (a) l og ( d U / dq ) a t q c T1T2T3 T4T5T6 (b)
T7T8T9 T10T11
2 4 6 (c) l og ( c ) a t q c T1T2T3 T4T5T6 (d)
T7T8T9 T10T11 -0.8-0.6-0.4
3 3.5 4 4.5 5 5.5 6 (e) l og ( m ) a t q c log( L ) T1T2 T3T4 T5T6
3 3.5 4 4.5 5 5.5 6 (f) log( L ) T7T8T9 T10T11
FIG. 4. (Color online) Derivative of Binder cumulant (top),magnetic susceptibility (middle), and magnetization (bottom)at the critical noise q c for the majority-vote model on theeleven Archimedean lattices as a function of linear clustersize L = √ N in log-log scale. The straight lines are linear fit,whose slopes represent critical exponents (1 /ν , γ/ν , and β/ν from top to bottom). They are listed in Table I. ical quantities. Note that similar suspicion was raised inRef. [35] in relation to the spin-1 Ising model. Wrongvalue of the critical noise must affect critical exponentsseriously.There are two ways to calculate critical exponents.Critical exponents ν and γ can be found from the max-imum values of physical quantities of Eqs. (6) and (8).This method is not efficient without extended ensemblemethods. The second method is to calculate the physicalquantities at the critical point. After the critical noise issettled, all the critical exponents can be obtained fromthe calculation only at the point. We used this methodand the results are shown in Fig. 4 and Table I. The lin-earity is remarkable. Large error bars for dU/dq are fromnumerical derivative of dU/dq = [ U ( q + δ ) − U ( q − δ )] / (2 δ ).This kind of calculation has substantial error inevitably,since small δ is required to reduce discretization error,but it increases the error in the numerator at the sametime. This problem becomes more serious especiallywhen the function has statistical error like this work.Therefore, 1 /ν has much larger uncertainty than γ/ν and β/ν .Another important source of error is from inaccuratecritical point. To estimate this kind of error, the criti-cal exponents γ/ν and β/ν are calculated assuming othervalues of critical noise close to the correct critical point.As shown in Fig. 5, values of critical exponents changea lot with varying the noise q : When q change by 0.1%, γ/ν and β/ν change by about 2% and 8%, respectively.For example, if we judge that β/ν should be obtainedwithin uncertainty of 8% to verify the universality class, (a) g / n (b) b / n ( q - q c ) / q c FIG. 5. Critical exponents γ/ν and β/ν as a function of noise q near the critical noise q c . These are obtained from fittingof magnetic susceptibility and magnetization at noise q . Theerror bars here do not take into account the uncertainty of q c .The critical exponents of the two-dimensional Ising universal-ity class are denoted by dashed horizontal lines ( γ/ν = 1 . β/ν = 1 . uncertainty of 0.1% is allowed at most for the criticalnoise. Since the critical noise in the square lattice is q c = 0 . . /ν , it has large uncertainty alreadyand it is relatively insensitive to the value of the criticalnoise.Scaling functions for susceptibility, magnetization, andBinder cumulant for different system sizes L are givenin Fig. 6. Only two cases are shown, but the other ninelattices show the same features. The critical exponentsobtained by this work were used. They all show clearscaling behavior confirming Eqs. (5), (6), and (7) andcritical exponents obtained in this work. We found thatvariation of critical exponents by a few percent does notchange the scaling plot noticeably. Therefore, accuratedetermination of critical exponents by scaling plot is very (a) T5 c L -g / n (b) T6 (c) m L b / n L =20 L =30 L =40 L =60 (d) L =20 L =30 L =40 L =60 0.0 0.2 0.4 0.6 -2 0 2 (e) U ( q - q c ) L n L = 80 L =120 L =160 L =200 -2 0 2 (f) ( q - q c ) L n L = 80 L =120 L =160 FIG. 6. Rescaled susceptibility ( χL − γ/ν ), magnetization( mL β/ν ), and Binder cumulant ( U ) as a function of rescalednoise [( q − q c ) L /ν ] for the trellis lattice (T5) and the Shastry-Sutherland lattice (T6). The critical exponents used in thisfigure are from Table I. difficult. IV. CONCLUSION
The MVM was studied on the eleven Archimedean lat-tices. The critical noises were calculated with unprece-dented accuracy and the critical exponents are obtainedbased on them. All the critical exponents are same asthose of the 2D Ising model within error bars. We con-clude that the MVM belongs to the Ising universalityclass at least within the Archimedean lattices. We alsoshowed that very accurate determination of the criticalnoise is required to calculate the critical exponents prop-erly.
ACKNOWLEDGMENTS
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