Systems with two symmetric absorbing states: Relating the microscopic dynamics with the macroscopic behavior

We propose a general approach to study spin models with two symmetric absorbing states. Starting from the microscopic dynamics on a square lattice, we derive a Langevin equation for the time evolution of the magnetization field, that successfully explains coarsening properties of a wide range of nonlinear voter models and systems with intermediate states. We find that the macroscopic behavior only depends on the first derivatives of the spin-flip probabilities. Moreover, an analysis of the mean-field term reveals the three types of transitions commonly observed in these systems—generalized voter, Ising and directed percolation. Monte Carlo simulations of the spin dynamics qualitatively agree with theoretical predictions.

Figure 1

(Color online) Flipping probability $f$ (left boxes) and its associated potential $V$ (right boxes) vs local field $\psi$. Curves correspond to coefficient values $a=-0.3$ (dotted), $a=0$ (solid), $a=0.3$ (dashed), for $b=-0.25$ (top) and $b=1.0$ (bottom). For both values of $b$, a single-well and double-well potentials are obtained for $a<0$ and $a>0$, respectively.

Figure 2

(Color online) Simulation results showing GV transitions on a 2D square lattice with first-nearest-neighbor interactions $(z=4)$. (a) $1∕\rho$ vs time on a log-linear scale, on a lattice of side $L=400$, starting from a random initial condition with zero magnetization. (b) Survival probability $P$ and (c) density of up spins $N$ on a ${800}^2$ lattice, starting from a quenched configuration. Curves are averages over 100 realizations in (a) and ${10}^5$ realizations in (b),(c), for $b=3.0$, 0.5, and $-0.25$ (from top to bottom) and several values of $a$ around the corresponding critical points $a_{\mathrm{GV}}=1.178$, 0.1, and $-0.1045$. Straight dotted lines have slopes $2∕\pi$ in (a), $-0.95$ in (b), and 0 in (c). Curves for $b=3.0$ and 0.5 are up-shifted for more clarity.

Figure 3

(Color online) GV, DP, and Ising transitions on a 2D square lattice of side $L=400$ with up to third-nearest-neighbor $(z=12)$. (a) $1∕\rho$ vs time on a log-linear scale, for $b=-0.25$ and values of $a$ around the GV critical point $a_{\mathrm{GV}}\simeq -0.1105$. (b) $\rho$ vs time on a log-log scale for $b=0.5$ and values of $a$ around the DP critical point $a_{\mathrm{DP}}\simeq 0.2127$. (c) Binder cumulant $U$ vs $a$ for $b=0.5$. Curves cross at $a_I\simeq 0.205$, where $U\simeq 0.56$, close to the universal value 0.6107 of the $d=2$ Ising model (horizontal dashed line). (a) Survival probability $P$ and (b) density of up spins $N$, on a ${800}^2$ lattice, starting from a quenched configuration. Curves are averages over ${10}^5$ realizations. Dashed straight lines have slopes $2∕\pi$ and $-1$ in (a) and its inset, respectively, whereas the slopes are $-0.45$ and 0.2295 in (b).