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arXiv:1507.05537 · Journal of Statistical Mechanics Theory and Experiment

Order–disorder transition in the two-dimensional interacting monomer-dimer model: Ising criticality

Su‐Chan Park

Published Oct 9, 2015• 5 citations
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Abstract

We study the order–disorder transition of the two-dimensional interacting monomer-dimer model (IMD) which has two symmetric absorbing states. To be self-contained, we first estimate numerically the dynamic exponent z of the two-dimensional Ising model. From the relaxation dynamics of the magnetization at the critical point, we obtain , or , where and are exactly known exponents. We then compare the critical relaxation of the order parameter at the transition point of the IMD with that of the Ising model. We find that the critical relaxation exponent is in good agreement with the Ising model, unlike the recent claim by Nam et al (2014 J. Stat. Mech . P08011). We also claim that the Binder cumulant is not an efficient quantity to locate the order–disorder transition point of the model with two symmetric absorbing states.

DOI
10.1088/1742-5468/2015/10/p10009

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We study the order–disorder transition of the two-dimensional interacting monomer-dimer model (IMD) which has two symmetric absorbing states. To be self-contained, we first estimate numerically the dynamic exponent z of the two-dimensional Ising model. From the relaxation dynamics of the magnetization at the critical point, we obtain , or , where and are exactly known exponents. We then compare the critical relaxation of the order parameter at the transition point of the IMD with that of the Ising model. We find that the critical relaxation exponent is in good agreement with the Ising model, unlike the recent claim by Nam et al (2014 J. Stat. Mech . P08011). We also claim that the Binder cumulant is not an efficient quantity to locate the order–disorder transition point of the model with two symmetric absorbing states.
We investigate the nonequilibrium relaxation dynamics of an interacting monomer-dimer model with nearest neighbor repulsion on a square lattice, which possesses two symmetric absorbing states. The model is known to exhibit two nearby continuous transitions: the Z 2 symmetry-breaking order-disorder transition and the absorbing transition with directed percolation criticality. We performed a more detailed analysis of our extensive simulations on bigger lattice systems which reaffirms that the symmetry-breaking transition exhibits a non-Ising critical behavior with β ≃ 0.149(2) and η ≃ 0.30(1) that are distinct from those values of a pure two dimensional Ising model. Finite size scaling of dimer density near the symmetry breaking transition gives logarithmic scaling ( α = 0.0) which is consistent with the hyperscaling relation but the corresponding exponent of ν B ≃ 1.37(2) exhibits a conspicuous deviation from the pure Ising value of 1. The value of dynamic critical exponent z , however, is found to be close to that of the kinetic Ising model as 1/ z ≃ 0.466(5) from the relaxation of staggered magnetization (and also similar but slightly smaller values from coarsening).
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.
We present a numerical study on an interacting monomer-dimer model with nearest neighbor repulsion on a square lattice, which possesses two symmetric absorbing states. The model is observed to exhibit two nearby continuous transitions: the Z2Z_2Z2​ symmetry-breaking order-disorder transition and the absorbing transition with directed percolation criticality. We find that the symmetry-breaking transition shows a non-Ising critical behavior, and that the absorbing phase becomes critical, in the sense that the critical decay of the dimer density observed at the absorbing transition persists even within the absorbing phase. Our findings call for further studies on the microscopic models and corresponding continuum description belonging to the generalized voter universality class.
We simulate the critical relaxation process of the two-dimensional Ising model with the initial state both completely disordered or completely ordered. Results of a method to measure both the dynamic and static critical exponents are reported, based on the finite-size scaling for the dynamics at the early time. From the time-dependent Binder cumulant, the dynamical exponent z is extracted independently, while the static exponents \ensuremath{\beta}/\ensuremath{\nu} and \ensuremath{\nu} are obtained from the time evolution of the magnetization and its higher moments. \textcopyright{} 1996 The American Physical Society.
We study the dynamic scaling behavior of a monomer-dimer model with repulsive itneractions between the same species in one dimension. With infinitely strong interactions the model exhibits a continuous transition from a reactive phase to an inactive phase with two equivalent absorbing states. This transition does not belong to the conventional directed percolation universality class. The values of dynamic scaling exponents are estimated by Monte Carlo simulations for two distinct initial configurations, one near an absorbing state and the other with an interface between two different absorbing states. We confirm that the critical behavior is consistent with that of the models with the mass conservation of modulo 2.
Abstract The critical behavior of a dimer model with an interaction favoring parallel dimers in each plaquette of the square lattice is studied numerically using the corner transfer matrix renormalization group algorithm. The critical exponents are known to depend on the chemical potential of vacancies, or monomers. At large average density of the latter, the phase transition becomes the first-order. We compute the scaling dimensions of both the order parameter and temperature in the second-order regime and compare them with the conjecture that the critical behavior is the same as the Ashkin–Teller model on its self-dual critical line.