Antiferromagnetic majority voter model on square and honeycomb lattices
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] S e p Antiferromagnetic majority voter model on square andhoneycomb lattices
Francisco Sastre a , Malte Henkel b a Departamento de Ingenier´a F´ısica, Universidad de Guanajuato, AP E-143, CP 37150,Le´on, M´exico b Groupe de Physique Statistique, Institut Jean Lamour (CNRS UMR 7198), Universit´e deLorraine Nancy, BP 70239, F – 54506 Vandœuvre-l`es-Nancy, France,
Abstract
An antiferromagnetic version of the well-known majority voter model on squareand honeycomb lattices is proposed. Monte Carlo simulations give evidencefor a continuous order-disorder phase transition in the stationary state in bothcases. Precise estimates of the critical point are found from the combinationof three cumulants, and our results are in good agreement with the reportedvalues of the equivalent ferromagnetic systems. The critical exponents 1 /ν , γ/ν and β/ν were found. Their values indicate that the stationary state of theantiferromagnetic majority voter model belongs to the Ising model universalityclass. Keywords:
Antiferromagnetic, Majority voter model, Finite-size scaling, Isingmodel, universality class
1. Introduction
The Majority voter model (MVM) is a simple non-equilibrium Ising-likesystem, proposed as a way to simulate opinion dynamics. The collective be-haviour of the voters shares many aspects with the well-established theory ofnon-equilibrium phase transitions and results form simulations can be analysedsimilarly [1]. In the standard MVM, the system evolves following a dynam-ics where each “voter” (spin) assumes the same opinion as the majority of itsneighbours, with probability (1 + x ) / − x ) /
2. Here x is the control parameter, with a range 0 ≤ x ≤ x c , and previous numerical works on regular lattices showthat the critical exponents are compatible with its equivalent lattices for theIsing model [2, 3, 4, 5]. Those results seem to confirm the conjecture thatnon-equilibrium models with up-down symmetry and spin-flip dynamics fall in Email addresses: [email protected] (Francisco Sastre), [email protected] (Malte Henkel)
Preprint submitted to Journal of L A TEX Templates August 15, 2018 he universality class of the Ising model [6]. However, numerical simulations innon-regular latices, such Archimedean, small-world or random lattices, ratherseem to indicate that the critical exponents are governed by the lattice topol-ogy [7, 8, 9, 10, 11, 12, 13, 14, 15].On the other hand, the MVM belongs to a family of generalized spin mod-els [16] that can be modelled in terms of a competing dynamics, induced by heatbaths at two different temperatures (on two dimensional square lattices) [17, 18]and hence have a non -equilibrium stationary state. The equilibrium stationarystate of the ferromagnetic Ising model is recovered when both temperaturesare equal; in this case, the detailed-balance condition is fulfilled. All membersof the family share the same definition of the order-parameter and there is acritical line that separates the paramagnetic (disordered) phase from the ferro-magnetic (ordered) phase [2]. So far, the ferromagnetic-paramagnetic transitionout of equilibrium has been extensively studied with a single-flip dynamic rulethat recovers the equilibrium Ising model. When antiferromagnetic interactionsare included in the Ising model, the phenomenology becomes richer, additionalphases with different critical behaviour, multicritical points, etc. can appear (seeRef. [19] and references therein). In this work, we want to explore, as a start-ing point, if it is possible to implement an antiferromagnetic non-equilibriumversion that follows a single-flip updating scheme. Godoy and Figureido [20]proposed a non-equilibrium mixed-spin antiferromagnetic model with two up-dating schemes: one single-spin via Glauber dynamics and a two-spin updatinglinked to an external energy source. The aim of the present work is to implementthe antiferromagnetic version of the MVM on the square and the honeycombtwo-dimensional lattices, evaluate the critical point and the stationary criticalexponents, β , γ and ν .This work is organised as follows: in section 2, the antiferromagnetic MVMis defined and the finite-size scaling method used to analyse its stationary stateis outlined. In section 3, the results of the Monte Carlo simulation are reportedand the critical parameters are extracted. We conclude in section 4.
2. Model and Finite-Size Scaling
In the ferromagnetic version of the MVM [2], each lattice site is occupied bya spin, σ i , that interacts with its nearest neighbours. The system evolves in thefollowing way: during an elementary time step, a spin σ i = ± p ( x ) = 12 (1 + ηx ) . (1)Herein, η stands for the (ferromagnetic) rule η = (cid:26) − sgn[ H i · σ i ] ; if H i = 00 ; if H i = 0 (2)and where H i is the local field produced by the nearest neighbours to the i thspin. The control parameter x acts analogously to a noise in the system. With2he dynamics (1,2), a given spin σ i adopts the sign of H i (the majority ofits nearest neighbours) with probability (1 + x ) / H i (the minority) with probability (1 − x ) /
2. In the Ising ferromagnet, the signof the bilinear exchange interaction in the hamiltonian defines if the system isferromagnetic, or antiferromagnetic, respectively, for a positive or negative sign.An anti ferromagnetic version of the MVM is now naturally obtained byreplacing the rule (2) by η = (cid:26) +sgn[ H i · σ i ] ; if H i = 00 ; if H i = 0 (3)in combination with the flip probability (1), with 0 < x <
1. The anti-ferroMVM is defined by the dynamics (1,3), with the control parameter 0 < x < x
7→ − x in the flipprobability (1). The consideration of square and honeycomb lattices allows tostudy the effect of having an even or odd number of nearest neighbours of eachspin on the critical behaviour in the stationary state.In analogy with simple Ising magnets, in order to measure the paramagnetic-antiferromagnetic phase transition, we shall use the staggered magnetisation asorder-parameter h m i = 1 N *(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X i c i σ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)+ , (4)where N is the total number of lattice sites, c i takes values of ± h . . . i stands for time averagetaken in the stationary regime. Figure 1: Sub-lattices for a) the square and b) the honeycomb geometries. Here, c i = 1for white sites and c i = − χ = N x {h m i − h m i } . (5)We shall use the method proposed in Ref. [21], where three different cumulantsare used for the evaluation of the critical point: (i) the fourth-order or Bindercumulant [22] U (4) = 1 − h m i h m i , (6)(ii) the third-order cumulant (where h m i is defined analogously to eq. (4)) U (3) = 1 − h m i h m ih m i , (7)and (iii) the second-order cumulant U (2) = 1 − h m i π h m i . (8)The scaling forms for the thermodynamic observables, in the stationary state,and together with the leading finite-size correction exponent ω are given by m ( ǫ, L ) ≈ L − β/ν ( ˆ M ( ǫL /ν ) + L − ω ˆˆ M ( ǫL )) , (9) χ ( ǫ, L ) ≈ L γ/ν ( ˆ χ ( ǫL /ν ) + L − ω ˆˆ χ ( ǫL )) , (10) U ( p ) ( ǫ, L ) ≈ ˆ U ( p ) ( ǫL /ν ) + L − ω ˆˆ U [ p ] ( ǫL ) . (11)where ǫ = x − x c is the distance from criticality, p = 2, 3 or 4, and L = √ N is thelinear size of the lattice. The parameters β , γ and ν are the critical exponentsfor the infinite system, see [1] for details.In principle, the critical point x c is found from the crossing points in thecumulants U ( p ) . A precise estimation of x c is achieved by taking into accountthe crossing points for different cumulants U ( p ) and U ( q ) with p = q arise fordifferent values of L . The values of x , where the cumulant curves U ( p ) ( x ) fortwo different linear sizes L i and L j intercept are denoted as x ( p ) ij . We expandEq. (11) around ǫ = 0 to obtain U ( p ) ≈ U ( p ) ∞ + ¯ U ( p ) ǫL /ν + ¯¯ U ( p ) L − ω + O( ǫ , ǫL − ω ) , (12)where U ( p ) ∞ are universal quantities, but ¯ U ( p ) and ¯¯ U ( p ) are non-universal. Thevalue of ǫ where the cumulant curves U ( p ) for two different linear sizes L i and L j intercept is denoted as ǫ ( p ) i,j . At this crossing point the following relation mustbe satisfied: L /νi ǫ ( p ) ij + B ( p ) L − ωi = L /νj ǫ ( p ) ij + B ( p ) L − ωj . (13)Here B ( p ) := ¯¯ U ( p ) / ¯ U ( p ) . Combining for different cumulants ( q = p ) we get x ( p ) ij + x ( q ) ij x c − ( x ( p ) ij − x ( q ) ij ) A pq , (14)4here A pq = ( B ( p ) + B ( q ) ) / [2( B ( p ) − B ( q ) )] and is non-universal (see Refs. [21,4] for additional details). Equation (14) is a linear equation that makes noreference to ν or ω and requires as inputs only the numerically measurablecrossing couplings x pi,j . The intercept with the ordinate gives the critical pointlocation.
3. Results
As a first step, we check that the dynamics Eqs. (1,3) gives rise to a pa- ramagnetic-antiferromagnetic phase transitions, depending on the value of theparameter x >
0. For a qualitative illustration, in Fig. 2 we present snapshotsfor two different values of x , of the state of a square lattice of size L = 256.Clearly, the two states look very different and are also distinguished by therespective values of h m i as defined in (4); this suggests the existence of a phasetransition, at some intermediate critical point x c . Figure 2: Snapshots of the square lattice with size L = 256, for a x = 0 . x = 0 . h m i ≃ . h m i ≃ . We performed simulations on three different lattices with linear sizes L = 24,28, 32, 36, 40 and 48; and did this for both the square and honeycomb lattices,see figure 1. Starting with a random configuration of spins, the system evolvesfollowing the dynamics given by eqs. (1,3). Then, we let the system evolve duringa transient time, that varied from 4 × Monte Carlo time steps (MCTS) for L = 24 to 1 . × MCTS for L = 48. Averages of the observables weretaken over 2 × MCTS for L = 24 and up to 1 . × MCTS for L = 48.Additionally, for each value of x and L , we performed up to 400 independentruns, in order to improve the statistics.In Fig. 3 we show the third-order cumulant curves as function of x for thesquare and for the honeycomb lattice. The figure shows clearly that in both ge-5metries the curves for different sizes cross around a certain (lattice-dependent)value of x . Similar behaviour has been observed as well for U (2) and for U (4) . x U ( ) L=24L=28L=32L=36L=40L=48 x U ( ) (a)(b) Figure 3: Third-order cumulant U (3) as function of the parameter x in (a) the square and (b)honeycomb lattices. The solid lines are the third-order polynomial fits. For the evaluation of the critical points, we used a third-order polynomialfit for the cumulant curves. Recalling eq. (14), the estimation of the criticalpoints for the two cases are shown in Figure 4, where we plot the variable σ := ( x (4) ij + x (2) ij ) / δ := x (4) ij − x (2) ij . We can observethat the raw differences between crosses, δ , are smaller in the square case. But,since the smaller values in δ are around 2 − for the square lattice and 2 − forthe honeycomb lattice (these values correspond to the crossings between thelargest sizes using in our simulation), we can be sure that largest sizes will notimprove significantly our results. In any case, the largest source of numericalerror apparently comes from the statistical uncertainty of the data points. Ifwe take another pair of cumulants, the definitions of σ and δ are adapted inan obvious way. The linear fits of Eq. (14) give the following estimates for thecritical points (cid:26) x c = 0 . x c = 0 . ×10 −5 δ σ ×10 −5 ×10 −4 δ σ (a)(b) Figure 4: (Color online) Evaluation of the critical point x c for (a) the square lattice and (b)the honeycomb lattice. The points represent the numerical data obtained from third-orderpolynomial fits and the dashed lines are the linear fits of Eq. (14). Smaller δ -values correspondto larger system sizes. Black circles: crossing of U (2) - U (4) , red squares: crossing of U (2) - U (3) ,blue diamonds: crossing of U (3) - U (4) . where the numbers in brackets give the estimated uncertainty in the last givendigit(s). Both results are in good agreement with the reported values for theferromagnetic MVM [2, 3, 5]. We want to point out that, when we compare thedata for the crossing of the U (2) − U (4) curves with the previous reported datafor the ferromagnetic case from Ref. [5], the range in the differences δ is almosttwo times larger in the ferromagnetic case. Since the linear sizes considered arethe same in both cases, we suspect that the scaling effects are smaller in theantiferromagnetic case. Additional simulations in the Ising model, for antiferro-magnetic and ferromagnetic, interactions for different lattice geometries and forthe MVM on square lattices would help to see if this is a universal behaviour.The critical exponents can be evaluated, by using Eqs. (11), at the critical7oint ǫ = 0. One expects the following finite-size scaling behaviour m ( L ) ∝ L − β/ν , (16) χ ( L ) ∝ L γ/ν , (17)and ∂U ( p ) ∂x (cid:12)(cid:12)(cid:12) x = x c ∝ L /ν , (18)In Fig. 5, we show the derivatives of the cumulants at the critical point. Fromthe finite-size scaling law (18), we obtain the following results: 1 /ν = 1 . /ν = 1 . U (2) , U (3) and U (4) respectively in the honeycomb case. The evaluation of γ/ν is shown in Fig. 6, with the relation (17) we obtain γ/ν = 1 . γ/ν = 1 . β/ν , with Eq. (16) we obtain β/ν = 0 . β/ν = 0 . β + γ ) /ν = d = 2, which is well confirmed by the simulation results. We alsosee that the agreement for the anti-ferromagnetic MVM is even slightly betterthan for the ferromagnetic MVM. It turned out that an explicit considerationof possible finite-size corrections, described by the Wegner exponent ω , was notnecessary. Table 1: Values of the critical parameters for the antiferromagnetic Majority voter model(AMVM). For comparison, we also include the values reported for the ferromagnetic Majorityvoter model (FMVM) by Kwak et al. [3] on square lattices and by Acu˜na-Lara et al. [5] onhoneycomb lattices. x c /ν γ/ν β/ν ( γ + 2 β ) /ν model0.84969(4) 1.02(4) 1.756(8) 0.123(2) 2.002(9) Square AMVM0.87195(14) 1.03(4) 1.759(10) 0.124(8) 2.007(15) Honeycomb AMVM0.8500(4) 0.98(3) 1.78(5) 0.120(5) 2.02(5) Square FMVM0.8721(1) 1.01(2) 1.755(8) 0.123(2) 2.001(9) Honeycomb FMVMAdditionally, we evaluated the universal quantities (lattice-dependent) U ( p ) ∞ for both lattices, using Eq. (11) with ǫ = 0. Our data do not allow to reliablyevaluate the Wegner exponent ω , since there is no need to include scaling cor-rections in these systems, as we illustrate in Fig. 8. Our result for the Bindercumulant, U (4) ∞ = 0 . U (4) = 0 . U (4) = 0 . ∂ U (2) ___ ∂ x ∂ U (3) ___ ∂ x
30 40 L ∂ U (4) ___ ∂ x (a)(b)(c) Figure 5: (Color online) Log-log plot of the derivatives of the cumulant (a) ∂U (2) /∂x , (b) ∂U (3) /partialx and (c) ∂U (4) /partialx , taken at the critical point x = x c . Black circlescorrespond to the square lattice and red squares to the honeycomb lattice. The dashed linesare power-law fits. both lattices, in Table 2. We observe that the results for the honeycomb andsquare lattices are clearly different from each other, in all cases. The expla-nation for this behaviour is simple: because in our simulations, the boundaryconditions of the two lattices are different: we use periodic boundary conditionsfor the square lattice and skew conditions in the horizontal boundaries for thehoneycomb lattice (see Fig. 1). Estimates of U (4) reported in the literature, andbased on simulations in two-dimensional critical Ising model, produced a simi-lar spread for different lattices or boundary conditions [25]. In order to test theuniversality through the cumulants U ( p ) , it would be necessary to perform sim-ulations for the Ising model on the honeycomb lattice with the same boundaryconditions. 9 L χ Figure 6: (Color online) Log-log plot of the susceptibility at the critical point for the square(black circles) and honeycomb (red squares) lattices. The dashed line are power-law fits.
4. Conclusions
We have introduced an antiferromagnetic version of the MVM, both on thetwo-dimensional square and honeycomb lattices, and studied its stationary state,as a function of the control parameter x , through intensive Monte Carlo simu-lations. The phase transition in the stationary state, between an ordered phasefor x large enough and a disordered phase for x small enough, falls on both lat-tices into the 2 D Ising model universality class, and independently of whetherthe number of nearest neighbours is even or odd. This model further illus-trates that the set of critical exponents is consistent in two-dimensional regularlattices. Future work should explore if additional phases and/or multicriticalpoints exist if an external field is included.10 L 〈 m 〉 Figure 7: (Color online) Log-log plot of the order parameter at the critical point for the square(black circles) and honeycomb (red squares) lattices. The dashed lines are power-law fits.
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