Phase transition in the majority-vote model on the Archimedean lattices

Phase transition in the majority-vote model on the Archimedean lattices


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 obtained with 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. It was shown that very precise determination of the critical noise is required to obtain proper values of the critical exponents.

05.20.-y, 05.70.Ln, 64.60.Cn, 05.50.+q

I Introduction

The Ising model Ising (1925) has played a crucial role in the development of important concepts in the statistical physics such as the phase transition, critical phenomena, and the universality class Brush (1967); McCoy and Maillard (2012). Although it was proposed to explain ferromagnetism, it can be applied to various phenomena of material systems: liquid-gas systems at the critical point, order-disorder transitions in binary alloy systems, charge-ordering in mixed-valence compounds, etc. In addition, recently, many kinds of varieties of the Ising model were proposed to explain social phenomena Stauffer (2008); Sen and Chakrabarti (2013). For example, spin state of a site in the Ising model may represent a social opinion or preference of a person, which can be affected by his or her acquaintances. An Ising-like model is also used to simulate racial segregation, where different spin states represent people of different races Schelling (1971). In this case, the total number of people of a race is constant and only the spacial configuration of people can change.

Compared with material systems, the social application of the Ising model should take into account two points. The first one is that the structure of social interactions are usually complex networks rather than periodic lattices Strogatz (2001). It changes the character of the phase transition of the Ising model Barrat and Weigt (2000); Aleksiejuk et al. (2002). The second point is that most of social phenomena are irreversible and out of equilibrium. Therefore, it is not trivial to apply the equilibrium statistical mechanics to the nonequilibrium social dynamics. Especially, it is not clear whether the universality hypothesis, which insists that systems of the same spacial dimension and the symmetry of the order parameter share the same critical exponents Stanley (1999), is applied also to nonequilibrium models.

Interestingly, it was proposed that the stochastic nonequilibrium systems with up-down symmetry belong to the equilibrium Ising universality class in a steady state Grinstein et al. (1985). If it is true, an Ising-like nonequilibrium model in any kind of two-dimensional (2D) lattice should have the same critical exponents as the equilibrium Ising model in the 2D square lattice Onsager (1944). It was confirmed in a few systems Tomé et al. (1991); de Oliveira et al. (1993); Ortega et al. (1998). As for the majority-vote model (MVM), which is one of the well-studied nonequilibrium models in the opinion dynamics with up-down symmetry, it was confirmed in the 2D square lattice de Oliveira (1992); Kwak et al. (2007). However, later a few other works on other 2D lattices (triangular, honeycomb, kagomé, maple-leaf, and bounce lattices) reported that critical exponents of the MVM are different from those of the Ising model Lima and Malarz (2006); Santos et al. (2011). More recently, it was argued again that the MVM on the triangular and honeycomb lattices belongs to the Ising universality class Acuña-Lara et al. (2014). In the three-dimensional simple-cubic lattice, there is also controversy about the universality class: Ref. [Yang et al., 2008] reported different critical exponents of the MVM from the Ising universality class, but Ref. [Acuña-Lara and Sastre, 2012] argued against that. Therefore, it is important to conclude about the nature of the phase transition of the MVM on the regular lattice.

In this work, phase transitions and critical phenomena of the MVM are studied on the eleven Archimedean lattices, which are 2D lattices by uniform tiling of regular polygons. It was proved that there exist only eleven Archimedean lattices Grünbaum and Shephard (1987): They are listed in Fig. 1 and Table 1. Due to simplicity and various topologies, they are good bases for systematic study on 2D systems Richter et al. (2004); Yu (2015a). The MVM on six Archimedean lattices has been studied de Oliveira (1992); Lima and Malarz (2006); Kwak et al. (2007); Santos et al. (2011); Acuña-Lara et al. (2014), and the results on the other five lattices are reported for the first time in this paper. By extensive Monte-Carlo calculations, the critical noise of each Archimedean lattice was obtained with unprecedented precision, and it is confirmed that the critical exponents of the MVM on all of the eleven Archimedean lattices are same as the 2D Ising universality class. Possible reasons for the discrepancies of previous works are also discussed.

Figure 1: The eleven Archimedean lattices.
Name     Ref.
T1 Triangular () 1 6 12 0.114(5) 1.08(6) 1.59(5), 1.64(1) 0.12(4) Santos et al. (2011)
0.1091(1) 1.01(2) 1.759(7) 0.123(2) Acuña-Lara et al. (2014)
0.10910(3) 1.01(5) 1.76(1) 0.125(3)
T2 Square () 1 4 8 0.075(1) 0.99(5) 1.70(8) 0.125(5) de Oliveira (1992)
0.075(1) 0.98(3) 1.78(5) 0.120(5) Kwak et al. (2007)
0.07518(3) 0.99(3) 1.75(1) 0.123(3)
T3 Honeycomb () 2 3 6 0.089(5) 0.87(5) 1.64(5), 1.66(8) 0.15(5) Santos et al. (2011)
0.0639(1) 1.01(2) 1.755(8) 0.123(2) Acuña-Lara et al. (2014)
0.06400(3) 1.01(4) 1.75(1) 0.122(3)
T4 Maple leaf () 6 5 9 0.134(3) 0.98(10) 1.632(35) 0.114(3) Lima and Malarz (2006)
0.09670(3) 0.99(6) 1.75(1) 0.124(3)
T5 Trellis () 2 5 10 0.10266(3) 1.01(6) 1.75(1) 0.125(3)
T6 Shastry-Sutherland () 4 5 11 0.10930(3) 1.04(6) 1.75(1) 0.125(3)
T7 Bounce () 6 4 8 0.091(2) 0.872(85) 1.596(54) 0.103(6) Lima and Malarz (2006)
0.05940(3) 0.99(4) 1.75(1) 0.123(3)
T8 Kagomé () 3 4 8 0.078(2) 0.86(6) 1.64(3), 1.62(5) 0.14(3) Santos et al. (2011)
0.06192(3) 1.03(5) 1.75(1) 0.125(3)
T9 Star () 6 3 4 0.00435(4) 1.02(6) 1.73(3) 0.124(5)
T10 SHD () 12 3 5 0.04282(3) 1.02(8) 1.75(1) 0.125(3)
T11 CaVO () 4 3 5 0.04925(3) 1.00(8) 1.75(1) 0.124(3)
2D Ising model (exact) 1 1.75 0.125 Onsager (1944)
Table 1: Name, number of lattice points per basis (), coordination number (), number of next-nearest neighbors (), critical noise (), critical exponents (, , and ), 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 mean square-hexagonal-dodecagonal and CaVO, respectively. Results of this work are in bold.

Ii Model and methods

The MVM used in this work is defined by the following spin flip probability de Oliveira (1992):


where the spin at site can have only and the summation is over nearest neighbors (NN) of site . The function is the sign function, which gives , , and zero for positive, negative, and zero , respectively. This update rule is a sort of death-birth dynamics, where one site is chosen to forget its spin state and its nearest neighbors determine a new spin state of the site Allen and Nowak (2014). In the MVM, it follows the spin of the majority with probability and that of the minority with probability . If the two kinds of neighbors tie, the spin is determined at random. The parameter is called the noise. For small , one of the two spin states will prevail the system, and the system will fluctuate randomly without order for close to half. It was shown that there exists a continuous phase transition at the critical noise between the two phases de Oliveira (1992). When is larger than half, the site prefers to choose the minority spin of nearest neighbors and another phase transition can exist Sastre and Henkel (2016).

In the case of equilibrium spin models, any Monte-Carlo dynamics that satisfies the detailed balance and the ergodicity gives equivalent results if the simulation is properly performed. However, nonequilibrium models depend on details of the update rule, and so cluster-update algorithms Swendsen and Wang (1987), which mitigate the critical slowing down, cannot be used. In addition, since there is no concept of energy and Boltzmann distribution Kwak et al. (2007), extended ensemble methods Iba (2001) also fail. Therefore, we performed the simulation directly using the update rule of Eq. (1). Most of the results were obtained by decreasing the noise very slowly from with random initial spin state, and the absence of hysteresis was verified. At each temperature, warming-up Monte-Carlo steps were followed by steps for measurement near the critical noise. Each site has a chance to flip its spin one time on average per one Monte-Carlo step. All the calculations were performed on parallelograms of size , with the number of sites per basis and the number of bases in one direction . The periodic boundary condition was used.

At each temperature, the magnetization , the magnetic susceptibility , and the fourth-order Binder cumulant are calculated as follows de Oliveira (1992).


In the definition of the susceptibility , sometimes a function of is multiplied Acuña-Lara and Sastre (2012); Acuña-Lara et al. (2014); Sastre and Henkel (2016); Kwak et al. (2007); Yang et al. (2008), but critical exponents are not affected by that because the functions do not show critical behavior at . Since energy and specific heat are not defined, only three critical exponents (, , and ) are defined Fisher and Barber (1972); Barber (1983):


where is the linear size of the cluster and , , and are scaling functions. The critical noise is determined first, and the critical exponents are obtained from the physical quantities (, , and ) calculated at de Oliveira (1992); Lima and Malarz (2006); Kwak et al. (2007); Yang et al. (2008); Santos et al. (2011); Acuña-Lara and Sastre (2012); Acuña-Lara et al. (2014).

Iii Results and Discussion

Figure 2: (Color online) Binder cumulant () as a function of noise () for the majority-vote model on the eleven Archimedean lattices with various linear sizes (). The critical noises () obtained by the crossing of the Binder cumulant are shown in vertical dashed lines.
Figure 3: (Color online) Magnetic susceptibility (), which is defined in Eq. (3), as a function of noise () for the majority-vote model on the eleven Archimedean lattices with various linear sizes (). The critical noises () obtained in Fig. 2 are represented by vertical solid lines.

The critical noise can be determined by the maximum of the magnetic susceptibility or the derivative of magnetization . The peak position depends on the linear cluster size by , where the parameter depends on the lattice type and the physical quantity measured Barber (1983). However, it is very difficult to locate the peak position precisely without extended ensemble methods Iba (2001). Alternatively, the critical noise can be found by the crossing of the fourth-order Binder cumulant, because the Binder cumulant becomes independent of the cluster size at by Eq. (7), ignoring the correction critical exponent () Yu (2015b). The error from the ignorance of the correction critical exponent is less than statistical error in this calculation. Figure 2 shows the Binder cumulant close to the critical point, and the critical noises obtained by this method are listed in Table 1. The magnetic susceptibility data in Fig. 3 support the results: The maximum susceptibility position approaches with increasing the cluster size and it is very close to for large clusters. As is expected, a lattice with more nearest neighbors tends to have higher critical noise, but differently from the Ising model, there are some exceptions Acuña-Lara et al. (2014). When the number of nearest neighbor is same, a lattice with more number of 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.)

Our results of the critical noise are consistent with Ref. de Oliveira (1992); Kwak et al. (2007); Acuña-Lara et al. (2014), but those of Ref. Lima and Malarz (2006); Santos et al. (2011) are much larger than ours out of error bars. They used smaller number of Monte-Carlo steps () for warming-up at each temperature, but we confirmed it is enough for moderate size clusters (). We performed the simulation with the same method explained in their papers, but failed to reproduce their results. Therefore, it would be reasonable to guess tentatively that there is an error in their code. It would be very difficult to find out the error, if there exists, without the code they used. However, the implementation of the lattice seems to be correct, because one of the authors could reproduce the exact values of Curie temperature for the ferromagnetic Ising model Codello (2010) within 1% in the maple-leaf and bounce lattices Malarz et al. (). Therefore, we guess something might be problematic in the majority-vote dynamics or in calculations of the physical quantities. Note that similar suspicion was raised in Ref. [Malakis et al., 2012] in relation to the spin-1 Ising model. Wrong value of the critical noise must affect critical exponents seriously.

Figure 4: (Color online) Derivative of Binder cumulant (top), magnetic susceptibility (middle), and magnetization (bottom) at the critical noise for the majority-vote model on the eleven Archimedean lattices as a function of linear cluster size in log-log scale. The straight lines are linear fit, whose slopes represent critical exponents (, , and from top to bottom). They are listed in Table 1.
Figure 5: Critical exponents and as a function of noise near the critical noise . These are obtained from fitting of magnetic susceptibility and magnetization at noise . The error bars here do not take into account the uncertainty of . The critical exponents of the two-dimensional Ising universality class are denoted by dashed horizontal lines ( and ).

There are two ways to calculate critical exponents. Critical exponents and can be found from the maximum values of physical quantities of Eqs. (6) and (8). This method is not efficient without extended ensemble methods. The second method is to calculate the physical quantities at the critical point. After the critical noise is settled, all the critical exponents can be obtained from the calculation only at the point. We used this method and the results are shown in Fig. 4 and Table 1. The linearity is remarkable. Large error bars for are from numerical derivative of . This kind of calculation has substantial error inevitably, since small is required to reduce discretization error, but it increases the error in the numerator at the same time. This problem becomes more serious especially when the function has statistical error like this work. Therefore, has much larger uncertainty than and .

Another important source of error is from inaccurate critical point. To estimate this kind of error, the critical exponents and are calculated assuming other values of critical noise close to the correct critical point. As shown in Fig. 5, values of critical exponents change a lot with varying the noise : When change by 0.1%, and change by about 2% and 8%, respectively. For example, if we judge that should be obtained within uncertainty of 8% to verify the universality class, uncertainty of 0.1% is allowed at most for the critical noise. Since the critical noise in the square lattice is , the maximum acceptable uncertainty is about , which is not satisfied by previous works de Oliveira (1992); Kwak et al. (2007). Thus, it can be argued that the uncertainty of critical exponents of all previous works are rather underestimated and their conclusions based on these calculations should be re-examined. Without correct and accurate calculation of the critical noise, accurate calculation of critical exponents is impossible and the decision about the universality class should be inconclusive. We estimate this is the source of a controversy about the universality class in this model. The uncertainty of the critical exponents of our work in Table 1 includes the effect of uncertainty of the critical noise. In the case of the critical exponent , it has large uncertainty already and it is relatively insensitive to the value of the critical noise.

Figure 6: Rescaled susceptibility (), magnetization (), and Binder cumulant () as a function of rescaled noise [] for the trellis lattice (T5) and the Shastry-Sutherland lattice (T6). The critical exponents used in this figure are from Table 1.

Scaling functions for susceptibility, magnetization, and Binder cumulant for different system sizes are given in Fig. 6. Only two cases are shown, but the other nine lattices show the same features. The critical exponents obtained by this work were used. They all show clear scaling behavior confirming Eqs. (5), (6), and (7) and critical exponents obtained in this work. We found that variation of critical exponents by a few percent does not change the scaling plot noticeably. Therefore, accurate determination of critical exponents by scaling plot is very difficult.

Iv Conclusion

The MVM was studied on the eleven Archimedean lattices. The critical noises were calculated with unprecedented accuracy and the critical exponents are obtained based on them. All the critical exponents are same as those of the 2D Ising model within error bars. We conclude that the MVM belongs to the Ising universality class at least within the Archimedean lattices. We also showed that very accurate determination of the critical noise is required to calculate the critical exponents properly.


This work was supported by the GIST Research Institute (GRI) in 2016.


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