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Physical Review A

Critical behavior of the three-dimensional contact process

Iwan Jensen

Published Jan 1, 1992• 58 citations
Publisher

Abstract

I determine the critical behavior of a nonequilibrium three-dimensional lattice model exhibiting a phase transition to an absorbing state. I study the model in the vicinity of the critical point, and in the subcritical region, via time-dependent Monte Carlo simulations. The method used in the subcritical region is very efficient. The results for the directly measured critical exponents, \ensuremath{\nu}=1.11\ifmmode\pm\else\textpm\fi{}0.01, \ensuremath{\eta}=0.114\ifmmode\pm\else\textpm\fi{}0.004, and z=1.052\ifmmode\pm\else\textpm\fi{}0.003, are consistent with those of directed percolation. \ensuremath{\delta}=0.732\ifmmode\pm\else\textpm\fi{}0.004 is obtained from the hyperscaling relation 4\ensuremath{\delta}+2\ensuremath{\eta}=dz, and \ensuremath{\beta}=0.813\ifmmode\pm\else\textpm\fi{}0.011 from \ensuremath{\beta}=\ensuremath{\nu}\ensuremath{\delta}. These results are the most precise so far for a three-dimensional model with directed percolation critical behavior.

DOI
10.1103/physreva.45.r563

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I determine the critical behavior of a nonequilibrium three-dimensional lattice model exhibiting a phase transition to an absorbing state. I study the model in the vicinity of the critical point, and in the subcritical region, via time-dependent Monte Carlo simulations. The method used in the subcritical region is very efficient. The results for the directly measured critical exponents, \ensuremath{\nu}=1.11\ifmmode\pm\else\textpm\fi{}0.01, \ensuremath{\eta}=0.114\ifmmode\pm\else\textpm\fi{}0.004, and z=1.052\ifmmode\pm\else\textpm\fi{}0.003, are consistent with those of directed percolation. \ensuremath{\delta}=0.732\ifmmode\pm\else\textpm\fi{}0.004 is obtained from the hyperscaling relation 4\ensuremath{\delta}+2\ensuremath{\eta}=dz, and \ensuremath{\beta}=0.813\ifmmode\pm\else\textpm\fi{}0.011 from \ensuremath{\beta}=\ensuremath{\nu}\ensuremath{\delta}. These results are the most precise so far for a three-dimensional model with directed percolation critical behavior.
I determine the critical behavior of a nonequilibrium three-dimensional lattice model exhibiting a phase transition to an absorbing state. Time-dependent Monte Carlo simulations in the vicinity of the critical point yield very accurate estimates for the location of the critical point and several critical exponents. Time-dependent simulations in the supercritical and subcritical regimes yield estimates for further critical exponents from the asymptotic behavior of the survival probability and the average number of particles. The results in the super- and subcritical regimes are also analyzed by utilizing the scaling laws describing the behavior of the model. The results for the critical exponents are the best obtained so far for a three-dimensional model with directed-percolation critical behavior.
The absorbing-state transition in the three-dimensional contact process with and without quenched randomness is investigated by means of Monte Carlo simulations. In the clean case, a reweighting technique is combined with a careful extrapolation of the data to infinite time to determine with high accuracy the critical behavior in the three-dimensional directed percolation universality class. In the presence of quenched spatial disorder, our data demonstrate that the absorbing-state transition is governed by an unconventional infinite-randomness critical point featuring activated dynamical scaling. The critical behavior of this transition does not depend on the disorder strength, i.e., it is universal. Close to the disordered critical point, the dynamics is characterized by the nonuniversal power laws typical of a Griffiths phase. We compare our findings to the results of other numerical methods, and we relate them to a general classification of phase transitions in disordered systems based on the rare region dimensionality.
I study a nonequilibrium lattice model, the pair contact process, in which pairs of particles annihilate with probability p or else create a particle at a vacant nearest neighbor. The model exhibits a continuous phase transition from an active state, with an ongoing production of particles, to an absorbing state without pairs. The model has infinitely many absorbing states. Computer simulations in 1D yielded critical exponents consistent with directed percolation, for the first time placing a model with infinitely many absorbing states firmly in the directed-percolation universality class.
In extensive Monte Carlo simulations the phase transition of the random-field Ising model in three dimensions is investigated. The values of the critical exponents are determined via finite-size scaling. For a Gaussian distribution of the random fields it is found that the correlation length \ensuremath{\xi} diverges with an exponent \ensuremath{\nu}=1.1\ifmmode\pm\else\textpm\fi{}0.2 at the critical temperature and that \ensuremath{\chi}\ensuremath{\sim}${\ensuremath{\xi}}^{2\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}}$ with \ensuremath{\eta}=0.50\ifmmode\pm\else\textpm\fi{}0.05 for the connected susceptibility and ${\mathrm{\ensuremath{\chi}}}_{\mathrm{dis}}$\ensuremath{\sim}${\ensuremath{\xi}}^{4\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}$ with \ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}=1.03\ifmmode\pm\else\textpm\fi{}0.05 for the disconnected susceptibility. Together with the amplitude ratio A=${\mathrm{lim}}_{\mathit{T}\ensuremath{\rightarrow}\mathit{T}\mathit{c}}$${\mathrm{\ensuremath{\chi}}}_{\mathrm{dis}}$/${\mathrm{\ensuremath{\chi}}}^{2}$(hr{\mathit{h}}_{\mathit{r}}hr​/T)2{)}^{2})2 being close to one this gives further support for a two-exponent scaling scenario implying \ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}=2\ensuremath{\eta}. The magnetization behaves discontinuously at the transition, i.e., \ensuremath{\beta}=0. However, no divergence for the specific heat and in particular no latent heat is found. Also the probability distribution of the magnetization does not show a multipeak structure that would be characteristic for the phase-coexistence at first-order phase-transition points.
The contact process is a paradigmatic classical stochastic system displaying critical behavior even in one dimension. It features a nonequilibrium phase transition into an absorbing state that has been widely investigated and shown to belong to the directed percolation universality class. When the same process is considered in a quantum setting, much less is known. So far, mainly semiclassical studies have been conducted and the nature of the transition in low dimensions is still a matter of debate. Also, from a numerical point of view, from which the system may look fairly simple-especially in one dimension-results are lacking. In particular, the presence of the absorbing state poses a substantial challenge, which appears to affect the reliability of algorithms targeting directly the steady state. Here we perform real-time numerical simulations of the open dynamics of the quantum contact process and shed light on the existence and on the nature of an absorbing state phase transition in one dimension. We find evidence for the transition being continuous and provide first estimates for the critical exponents. Beyond the conceptual interest, the simplicity of the quantum contact process makes it an ideal benchmark problem for scrutinizing numerical methods for open quantum nonequilibrium systems.