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arXiv:cond-mat/0010045 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

One-dimensional contact process: Duality and renormalization

Jef Hooyberghs, Carlo Vanderzande

Published Mar 22, 2001• 18 citations
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Abstract

We study the one-dimensional contact process in its quantum version using a recently proposed real-space renormalization technique for stochastic many-particle systems. Exploiting the duality and other properties of the model, we can apply the method for cells with up to 37 sites. After suitable extrapolation, we obtain exponent estimates that are comparable in accuracy with the best known in the literature.

DOI
10.1103/physreve.63.041109

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We study the one-dimensional contact process in its quantum version using a recently proposed real-space renormalization technique for stochastic many-particle systems. Exploiting the duality and other properties of the model, we can apply the method for cells with up to 37 sites. After suitable extrapolation, we obtain exponent estimates that are comparable in accuracy with the best known in the literature.
The stationary state of stochastic processes such as reaction-diffusion systems can be related to the ground state of a suitably defined quantum Hamiltonian. Using this analogy, we investigate the applicability of a real-space renormalization group approach, originally developed for quantum spin systems, to interacting particle systems. We apply the technique to an exactly solvable reaction-diffusion system and to the contact process (both in d = 1). In the former case, several exact results are recovered. For the contact process, surprisingly good estimates of critical parameters are obtained from a small-cell renormalization.
In the theory of stochastic many particle systems one models phenomena of non-equilibrium statistical mechanics by means of a Markov process. The master equation, describing the time evolution of such a process, can formally be interpreted as a Schrodinger equation in imaginary time. In this approach, the (non-Hermitian) generator of the process plays the role of the Hamiltonian. This mathematical equivalence permits the successful application to stochastic systems of a number of exact, approximative or numerical methods from quantum mechanics.A particularly interesting type of stochastic systems is formed by the processes having a phase transition between an absorbing state and an active state. Examples of these systems can be found in physics, chemistry and even biology. The best known process of this kind is the contact process. In this thesis real-space renormalisation group techniques, which were originally developed for quantum mechanical systems, are applied to study stochastic models on a one-dimensional lattice, more precisely the critical behaviour and the universality of absorbing state phase transitions. First, the standard renormalisation group (SRG) is adapted to a stochastic context resulting in one of the few analytical techniques capable of producing approximations for systems that can not be solved exactly. The SRG is successfully applied to the contact process for which it characterises the stationary critical behaviour with a competitive accuracy. Next a generalisation of the contact process to a model with several absorbing states is studied by means of the density matrix renormalisation group (DMRG) algorithm. The results for the cases with one and two absorbing states are consistent with what was known from simulations done by an other author. For the models with more than two absorbing states new results were found, which are supported by analytical considerations in a particular limit. Finally the experience of the previous two projects is used in the investigation of the quenched random contact process. In this model the local interactions on a lattice site are time-invariant stochastic variables, independently drawn from a given distribution with a certain variance, which gives a measure for the disorder. Very little was known about the stationary behaviour of this model since its extremely slow dynamics makes simulations inefficient. The renormalisation techniques in this thesis, suffer less from this problem. First, for the case of small disorder the SRG indicates that the critical behaviour is changed by the presence of disorder and the cross-over exponent is consistent with the Harris criterion. For intermediate disorder a DMRG study reveals that the critical exponents are continuously varying with the amount of disorder. Finally, an SRG calculation, suitably adapted to strong disorder, shows that in this case the phase transition is characterised by a fixed point at infinite randomness. In this limit the renormalisation is expected to be exact. It is the first time exact critical exponents are found for a phase transition out of an absorbing state. Moreover, the exponents are the same as those previously found in a class of disordered quantum chains, suggesting a new kind of universality for strongly disordered systems.
The stationary state of stochastic processes such as reaction-diffusion systems can be related to the ground state of a suitably defined quantum Hamiltonian. Using this analogy, we investigate the applicability of a real space renormalisation group approach, originally developped for quantum spin systems, to interacting particle systems. We apply the technique to an exactly solvable reaction-diffusion system and to the contact process (both in d=1d=1d=1). In the former case, several exact results are recovered. For the contact process, surprisingly good estimates of critical parameters are obtained from a small-cell renormalisation.
Sarker's real-space renormalization-group method, applied previously to the one-dimensional dimerized XY model, is used to study the one-dimensional pure and dimerized Heisenberg models. Comparison of critical exponents (the dynamical and the gap exponents) with exact or conjectured values indicates that the method, in its original version, does not yield an improved real-space technique, at least for the models studied for which the opening of the gap leads to essential singularity or logarithmic corrections.
Two different real-space techniques, the renormalization-group blocking method and finitecell scaling, are applied to a Hamiltonian of interacting fermions in one dimension which considers repulsion between electrons on neighboring sites. The localization transition which occurs at T=0T=0T=0 when the strength of the electron repulsion increases is qualitatively well described by both techniques. However, the essential singularity which appears in the exact solution is only recovered by the finite-cell-scaling method.