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arXiv:2002.03143 · Physical review. E

Order-parameter critical exponent of absorbing phase transitions in one-dimensional systems with two symmetric absorbing states

Su‐Chan Park

Published May 14, 2020• 2 citations
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Abstract

Via extensive Monte Carlo simulations along with systematic analyses of corrections to scaling, we estimate the order parameter critical exponent β of absorbing phase transitions in systems with two symmetric absorbing states. The value of β was conjectured to be 13/14≈0.93 and Monte Carlo simulation studies in the literature have repeatedly reproduced values consistent with the conjecture. In this paper, we systematically estimate β by analyzing the effective exponent after finding how strong corrections to scaling are. We show that the widely accepted numerical value of β is not correct. Rather, we obtain β=1.020(5) from different models with two symmetric absorbing states.

DOI
10.1103/physreve.101.052114

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Via extensive Monte Carlo simulations along with systematic analyses of corrections to scaling, we estimate the order parameter critical exponent β of absorbing phase transitions in systems with two symmetric absorbing states. The value of β was conjectured to be 13/14≈0.93 and Monte Carlo simulation studies in the literature have repeatedly reproduced values consistent with the conjecture. In this paper, we systematically estimate β by analyzing the effective exponent after finding how strong corrections to scaling are. We show that the widely accepted numerical value of β is not correct. Rather, we obtain β=1.020(5) from different models with two symmetric absorbing states.
Via extensive Monte Carlo simulations along with systematic analyses of corrections to scaling, we estimate the order parameter critical exponent β\betaβ of the directed Ising universality class. The value of β\betaβ was conjectured to be 1314\frac{13}{14}1413​ and Monte Carlo simulation studies in the literature have repeatedly reproduced the consistent values to the conjecture. In this paper, however, we find that the conjectured value of β\betaβ is not correct. Rather, we obtain β\betaβ = 1.020(5) from different models belonging to the directed Ising class.
Recently, Takayasu and Tretyakov [Phys. Rev. Lett. 68, 3060 (1992)] studied branching annihilating random walks with n=1-5 offsprings. These models exhibit a continuous phase transition to an absorbing state. Steady-state simulations yielded an estimate for the order parameter critical exponent \ensuremath{\beta} different from that of directed percolation. This result is quite surprising, as the universality class of directed percolation is known to be very robust. I have studied the critical behavior of the one-dimensional model with n=1 and 3 using time-dependent Monte Carlo simulations, and determined three critical exponents, all of which are in agreement with directed percolation.
We study a one-dimensional model that undergoes a transition between an active and an absorbing phase. Monte Carlo simulations supported by some additional arguments prompted us to predict the exact location of the critical point and critical exponents in this model. The exponents delta=0.5 and z=2 follows from random-walk-type arguments. The exponents beta=nu( perpendicular) are found to be nonuniversal and encoded in the singular part of reactivation probability, as recently discussed by H. Hinrichsen (cond-mat/0008179). A related model with quenched randomness is also studied.
We study a model of two-dimensional interacting monomers which has two symmetric absorbing states and exhibits two kinds of phase transition; one is an order-disorder transition and the other is an absorbing phase transition. Our focus is around the order-disorder transition, and we investigate whether this transition is described by the critical exponents of the two-dimensional Ising model. By analyzing the relaxation dynamics of "staggered magnetization," the finite-size scaling, and the behavior of the magnetization in the presence of a symmetry-breaking field, we show that this model should belong to the Ising universality class. Our results along with the universality hypothesis support the idea that the order-disorder transition in two-dimensional models with two symmetric absorbing states is of the Ising universality class, contrary to the recent claim [K. Nam et al., J. Stat. Mech.: Theory Exp. (2011) L06001]. Furthermore, we illustrate that the Binder cumulant could be a misleading guide to the critical point in these systems.
With extensive Monte Carlo simulations, we present high-precision estimates of the critical exponents of branching annihilating random walks with two offspring, a prototypical model of the directed Ising universality class in one dimension. To estimate the exponents accurately, we propose a systematic method to find corrections to scaling whose leading behavior is supposed to take the form t −χ in the long-time limit at the critical point. Our study shows that χ ≈ 0.75 for the number of particles in defect simulations and χ ≈ 0.5 for other measured quantities, which should be compared with the widely used value of χ = 1. Using χ so obtained, we analyze the effective exponents to find that β/ν ‖ = 0.2872(2), z = 1.7415(5), η = 0.0000(2), and accordingly, β/ν ⊥ = 0.5000(6). Our numerical results for β/ν ‖ and z are clearly different from the conjectured rational numbers β/ν ‖ = $\tfrac{2} {7} $ ≈ 0.2857, z = $\tfrac{7} {4} $ = 1.75 by Jensen [Phys. Rev. E, 50, 3623 (1994)]. Our result for β/ν ⊥, however, is consistent with $\tfrac{1} {2} $ , which is believed to be exact.
We study the critical behavior of the absorbing-state phase transition depicted by a contact process one-dimensional model system with power-law decaying interactions. The system dynamical processes include particle creation at a rate which decays with the distance from the nearest particle as $1/{r}^{\ensuremath{\alpha}}$. This model displays an absorbing-state phase transition with critical exponents varying continuously with the interaction exponent $\ensuremath{\alpha}$. Here, we provide a finite-size scaling analysis of the stationary order-parameter density, one of its moment ratio, its logarithmic derivative, and fluctuations. We also follow the short-time relaxation dynamic of these quantities to estimate their corresponding dynamical critical exponents. The estimated exponents are shown to be consistent with the hyperscaling relation. Further, we report an unconventional regime on which the critical order-parameter fluctuations vanish in the thermodynamic limit.