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Physical Review Letters

Critical Behavior of the Contact Process with Parity Conservation

Norio Inui, A. Yu. Tretyakov

Published Jun 8, 1998• 17 citations
Publisher

Abstract

We introduce a parity conserving version of the contact process, which can be treated using a series expansion technique. Both Pad\'e approximants for the series and numerical simulations for the derivative of the order parameter show that the critical exponent $\ensuremath{\beta}$ is consistent with the conjecture $\ensuremath{\beta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$. Dynamical Monte Carlo simulations confirm that the model belongs to the branching annihilating random walk with an even number of offsprings universality class.

DOI
10.1103/physrevlett.80.5148

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We introduce a parity conserving version of the contact process, which can be treated using a series expansion technique. Both Pad'e approximants for the series and numerical simulations for the derivative of the order parameter show that the critical exponent \ensuremathβ\ensuremath{\beta}\ensuremathβ is consistent with the conjecture \ensuremathβ=1\ensuremath{\beta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1\ensuremathβ=1. Dynamical Monte Carlo simulations confirm that the model belongs to the branching annihilating random walk with an even number of offsprings universality class.
In the preceding article Carlon et al. investigate the critical behavior of the pair contact process with diffusion. Using density matrix renormalization group methods, they estimate the critical exponents, raising the possibility that the transition might belong to the same universality class as branching annihilating random walks with even numbers of offspring. This is surprising since the model does not have an explicit parity-conserving symmetry. In order to understand this contradiction, we estimate the critical exponents by Monte Carlo simulations. The results suggest that the transition might belong to a different universality class that has not been investigated before.
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.
In this dissertation we address the problem of second order phase transitions with absorbing states. We analyze the critical behavior of a one dimensional model simulated on one dimensional chains. The sites on the linear chain can be multiple occupied by particles that annhilates each other with a finite probability. It also presents parity-conserving branching with an even number of offsprings. The number of particles are constant modulo 2. We determine the threshold of the phase transition between the statistically stationary active state and the absorbing state. From steady-state simulations and a finite-size scaling analysis, we calculate the order-parameter, order-parameter fluctuations, and special correlation length critical exponents. We also followed the short-time critical relaxation to determine some dynamical critical exponents. The hyperscaling relation is checked for both sets of stationary and dynamical critical exponents which we found to be consistent with the BARW-PC universality class.
Recently it was suggested that a pair contact process with diffusion (PCPD) might represent an independent new universality class different from the directed percolation (DP) and the parity conservation (PC) class. The dynamics in the PCPD are usually controlled by two independent parameters. The critical exponents for the PCPD are known to have different values for varying values of the two independent parameters. However, once the diffusion and annihilation (or coagulation) rate in the PCPD is tuned in a way that the process without offspring production is exactly solvable, a well-defined set of the exponents for the PCPD is obtained. Then dynamics are controlled by only one independent parameter. The obtained critical exponents are different than those of DP and PC. The critical exponents satisfy the generalized hyperscaling relation within numerical errors.
We provide finite-size scaling estimates for the dynamical critical exponent of the even parity-conserving universality class of critical behavior through exact numerical diagonalizations of the time evolution operator of an even-parity-conserving contact process. Our data seem to indicate that upon the introduction of a small diffusion rate in the process its critical behavior crosses over to that of the directed percolation universality class. A brief discussion of the many-sector decomposability of parity-conserving contact processes is presented.