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arXiv:1612.02999 · Physical review. E

Mixed-order phase transition of the contact process near multiple junctions

Róbert Juhász, Ferenc Iglói

Published Feb 7, 2017• 18 citations
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Abstract

We have studied the phase transition of the contact process near a multiple junction of M semi-infinite chains by Monte Carlo simulations. As opposed to the continuous transitions of the translationally invariant (M=2) and semi-infinite (M=1) system, the local order parameter is found to be discontinuous for M>2. Furthermore, the temporal correlation length diverges algebraically as the critical point is approached, but with different exponents on the two sides of the transition. In the active phase, the estimate is compatible with the bulk value, while in the inactive phase it exceeds the bulk value and increases with M. The unusual local critical behavior is explained by a scaling theory with an irrelevant variable, which becomes dangerous in the inactive phase. Quenched spatial disorder is found to make the transition continuous in agreement with earlier renormalization group results.

DOI
10.1103/physreve.95.022109

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We have studied the phase transition of the contact process near a multiple junction of M semi-infinite chains by Monte Carlo simulations. As opposed to the continuous transitions of the translationally invariant (M=2) and semi-infinite (M=1) system, the local order parameter is found to be discontinuous for M>2. Furthermore, the temporal correlation length diverges algebraically as the critical point is approached, but with different exponents on the two sides of the transition. In the active phase, the estimate is compatible with the bulk value, while in the inactive phase it exceeds the bulk value and increases with M. The unusual local critical behavior is explained by a scaling theory with an irrelevant variable, which becomes dangerous in the inactive phase. Quenched spatial disorder is found to make the transition continuous in agreement with earlier renormalization group results.
We consider the contact process near an extended surface defect, where the local control parameter deviates from the bulk one by an amount of λ(l)-λ(∞)=Al^{-s}, with l being the distance from the surface. We concentrate on the marginal situation s=1/ν_{⊥}, where ν_{⊥} is the critical exponent of the spatial correlation length, and study the local critical properties of the one-dimensional model by Monte Carlo simulations. The system exhibits a rich surface critical behavior. For weaker local activation rates A<A_{c}, the phase transition is continuous, having an order-parameter critical exponent, which varies continuously with A. For stronger local activation rates A>A_{c}, the phase transition is of mixed order: the surface order parameter is discontinuous; at the same time the temporal correlation length diverges algebraically as the critical point is approached, but with different exponents on the two sides of the transition. The mixed-order transition regime is analogous to that observed recently at a multiple junction and can be explained by the same type of scaling theory.
We study two models having an infinite-disorder critical point—the zero temperature random transverse-field Ising model and the random contact process—on a star-like network composed of M semi-infinite chains connected to a common central site. By the strong disorder renormalization group method, the scaling dimension x M of the local order parameter at the junction is calculated. It is found to decrease rapidly with the number M of arms, but remains positive for any finite M . This means that, in contrast with the pure transverse-field Ising model, where the transition becomes of first order for M > 2, it remains continuous in the disordered models, although, for not too small M , it is hardly distinguishable from a discontinuous one owing to a close-to-zero x M . The scaling behavior of the order parameter in the Griffiths–McCoy phase is also analyzed.
We study two models having an infinite-disorder critical point — the zero temperature random transverse-field Ising model and the random contact process — on a star-like network composed of M semi-infinite chains connected to a common central site. By the strong disorder renormalization group method, the scaling dimension xM of the local order parameter at the junction is calculated. It is found to decrease rapidly with the number M of arms, but remains positive for any finite M . This means that, in contrast with the pure transverse-field Ising model, where the transition becomes of first order for M > 2, it remains continuous in the disordered models, although, for not too small M , it is hardly distinguishable from a discontinuous one owing to a close-to-zero xM . The scaling behavior of the order parameter in the Griffiths-McCoy phase is also analyzed.
We investigate the nonequilibrium phase transition in the disordered contact process in the presence of long-range spatial disorder correlations. These correlations greatly increase the probability for finding rare regions that are locally in the active phase while the bulk system is still in the inactive phase. Specifically, if the correlations decay as a power of the distance, the rare-region probability is a stretched exponential of the rare-region size rather than a simple exponential as is the case for uncorrelated disorder. As a result, the Griffiths singularities are enhanced and take a non-power-law form. The critical point itself is of infinite-randomness type but with critical exponent values that differ from the uncorrelated case. We report large-scale Monte Carlo simulations that verify and illustrate our theory. We also discuss generalizations to higher dimensions and applications to other systems such as the random transverse-field Ising model, itinerant magnets, and the superconductor-metal transition.
We investigate the influence of time-varying environmental noise, i.e., temporal disorder, on the nonequilibrium phase transition of the contact process. Combining a real-time renormalization group, scaling theory, and large scale Monte-Carlo simulations in one and two dimensions, we show that the temporal disorder gives rise to an exotic critical point. At criticality, the effective noise amplitude diverges with increasing time scale, and the probability distribution of the density becomes infinitely broad, even on a logarithmic scale. Moreover, the average density and survival probability decay only logarithmically with time. This infinite-noise critical behavior can be understood as the temporal counterpart of infinite-randomness critical behavior in spatially disordered systems, but with exchanged roles of space and time. We also analyze the generality of our results, and we discuss potential experiments.
We study the nonequilibrium phase transition in the one-dimensional contact process with quenched spatial disorder by means of large-scale Monte Carlo simulations for times up to 10(9) and system sizes up to 10(7) sites. In agreement with recent predictions of an infinite-randomness fixed point, our simulations demonstrate activated (exponential) dynamical scaling at the critical point. The critical behavior turns out to be universal, even for weak disorder. However, the approach to this asymptotic behavior is extremely slow, with crossover times of the order of 10(4) or larger. In the Griffiths region between the clean and the dirty critical points, we find power-law dynamical behavior with continuously varying exponents. We discuss the generality of our findings and relate them to a broader theory of rare region effects at phase transitions with quenched disorder.