Majority-vote model on hyperbolic lattices

We study the critical properties of a nonequilibrium statistical model, the majority-vote model, on heptagonal and dual heptagonal lattices. Such lattices have the special feature that they only can be embedded in negatively curved surfaces. We find, by using Monte Carlo simulations and finite-size analysis, that the critical exponents $1/\nu$, $\beta /\nu$, and $\gamma /\nu$ are different from those of the majority-vote model on regular lattices with periodic boundary condition, which belongs to the same universality class as the equilibrium Ising model. The exponents are also from those of the Ising model on a hyperbolic lattice. We argue that the disagreement is caused by the effective dimensionality of the hyperbolic lattices. By comparative studies, we find that the critical exponents of the majority-vote model on hyperbolic lattices satisfy the hyperscaling relation $2\beta /\nu +\gamma /\nu =D_{\text{eff}}$, where $D_{\text{eff}}$ is an effective dimension of the lattice. We also investigate the effect of boundary nodes on the ordering process of the model.

Figure 1

A heptagonal lattice with level 3 (left) and the dual lattice with level 4 (right). For the heptagonal lattices with level $l=2,3,4,5,6,7$, the total number of nodes are 35, 112, 315, 847, 2240, and 5887, respectively. For the dual lattice with level $l=3,4,5,6,7,8$, the total number of nodes are 29, 85, 232, 617, 1625, and 4264, respectively.

Figure 2

(Color online) Majority-vote model on a two dimensional square lattice with a free boundary condition. (a) Magnetization $M$, (b) susceptibility $\chi$, and (c) reduced fourth-order cumulant $U$ as a function of the noise parameter $q$ for several values of the system size $N$. In (c), within the accuracy of the data, all curves intersect at $q_c=0.074\left(5\right)$.

Figure 3

(Color online) (a) Data collapse of the magnetization $M$ shown in Fig. 2a. The exponents used for the data collapse are $1/\nu =0.5$ and $\beta /\nu =0.0625$. (b) Log-log plot of the maximum of the susceptibility as a function of $N$. From it we estimate the critical exponent $\gamma /\nu =0.824\left(7\right)$ as the best fit of the data points. (c) Data collapse of the susceptibility shown in Fig. 2c. The fitting exponents are $\beta /\nu =0.0625$ and $\beta /\nu =0.825$.

Figure 4

(Color online) Majority-vote model on the heptagonal lattices. (a) Magnetization $M$, (b) susceptibility $\chi$, and (c) reduced fourth-order cumulant $U$, as a function of the noise parameter $q$ for several values of the system size $N$. The critical point $q_c=0.034\left(8\right)$ is estimated as the point at which the different curves for different $N$ intercept each other.

Figure 5

(Color online) (a) Log-log plot of the size dependence of the maximum values of derivatives of various thermodynamic quantities used to determine $1/\nu$. (b)Log-log plot of the magnetization at $q=q_c$ as a function of $N$. The slope of the best fit gives $\beta /\nu =0.114\left(5\right)$. (c) Data collapse of the magnetization $M$ shown in Fig. 4a. The exponents used for the data collapse $1/\nu =0.3$ and $\beta /\nu =0.115$. (d) Log-log plot of the maximum of the susceptibility as a function of $N$. From this we estimate the critical exponent $\gamma /\nu =0.721\left(7\right)$ as the best fit of data points. (e) Data collapse of the susceptibility shown in Fig. 4c. The fitting exponents $\beta /\nu =0.722$ and $1/\nu =0.3$.

Figure 6

(Color online) Majority-vote model on the dual heptagonal lattices. (a) Magnetization $M$, (b) susceptibility $\chi$, and (c) reduced fourth-order cumulant $U$, as a function of the noise parameter $q$ for several values of the system size $N$. The critical point $q_c=0.087\left(5\right)$ is estimated as the point where the curves for different $N$ intercept.

Figure 7

(Color online) (a) Log-log plot of the size dependence of the maximum values of derivatives of various thermodynamic quantities used to determine $1/\nu$. (b) Log-log plot of the magnetization at $q=q_c$ as a function of $N$. The slope of the best fit of the points gives $\beta /\nu =0.093\left(6\right)$. (c) Data collapse of the magnetization $M$ shown in Fig. 6a. The exponents used for the data collapse $1/\nu =1/3$ and $\beta /\nu =0.094$. (d) Log-log plot of the maximum of the susceptibility as a function of $N$. From this plot we estimate the critical exponent $\gamma /\nu =0.761\left(9\right)$. (e) Data collapse of the susceptibility shown in Fig. 6c. The fitting exponents $1/\nu =1/3$ and $\beta /\nu =0.762$.

Figure 8

(Color online) Magnetization $M$ as a function of the noise parameter $q$. (a) Filled squares represent the magnetization at the inner level 3 of the heptagonal lattice with seven levels in total. Open circles represent the heptagonal lattice with three levels. (b), (c), and (d) show the same situation as (a) but for different levels.

Figure 9

(Color online) Magnetization $M$ as a function of the noise parameter $q$. (a) Filled squares symbolize the magnetization in the inner level 4 of the dual heptagonal lattice with eight levels in total. Open circles represent the heptagonal lattice with four levels. (b), (c), and (d) show plots corresponding to (a) for other levels.