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arXiv:1711.03495 · Physical review. E

Nonuniversal and anomalous critical behavior of the contact process near an extended defect

Róbert Juhász, Ferenc Iglói

Published Jan 10, 2018• 2 citations
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Abstract

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.

DOI
10.1103/physreve.97.012111

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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 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 study the local critical behavior near the free surface of a d-dimensional system with a smooth inhomogeneity induced by a defect of dimension d∗{\mathit{d}}^{\mathrm{*}}d∗\ensuremath{\le}d-1 located on the surface. The couplings deviate from their bulk critical value by \ensuremath{\lambda}${\mathit{r}}_{\mathrm{\ensuremath{\perp}}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\alpha}}}$ where ${\mathit{r}}_{\mathrm{\ensuremath{\perp}}}$ is the distance to the center of the defect. Using a phenomenological approach, the change in the surface critical exponents is calculated to first order in the perturbation amplitude \ensuremath{\lambda} when the system is marginal, i.e., when the decay exponent of the perturbation \ensuremath{\alpha} is equal to the scaling dimension ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{\ensuremath{\varepsilon}}}^{0}$ of the bulk energy-density operator. Finally, the critical behavior near extended defects in the bulk is compared to the surface case.
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.
One of the most striking aspects of critical behaviour is that of the crucial role played by the geometry of the System. Critical exponents depend in a non-trivial manner on the dimensionality d . The very existence of a phase transition depends on the way in which the infinite volume limit is taken, as discussed in Section 4.4. This happens because the critical fluctuations, which determine the universal properties, occur at long wavelengths and are therefore very sensitive to the large scale geometry. By contrast, the non-critical properties of a system are sensitive to fluctuations on the scale of the correlation length and are therefore much less influenced. This line of reasoning also suggests that not all points in a system are equivalent in the way the local degrees of freedom couple to these critical fluctuations. So far, we have considered the behaviour of scaling operators only at points deep inside the bulk of a system. Near a boundary, however, the local environment of a given degree of freedom is different, and we might expect to find different critical properties there. In general, such differences should extend into the bulk only over distances of the order of the bulk correlation length. However, at a continuous bulk phase transition, this distance diverges, and we should expect the influence of boundaries to be more pronounced. The simplest modification of the bulk geometry to consider is that of a ( d − 1)-dimensional hyperplane bounding a semi-infinite d -dimensional system.
The theory of interface localization in near-critical planar systems at phase coexistence is formulated from first principles. We show that mutual delocalization of two interfaces, amounting to interfacial wetting, occurs when the bulk correlation length critical exponent ν is larger than or equal to 1. Interaction with a boundary or defect line involves an additional scale and a dependence of the localization strength on the distance from criticality. The implications are particularly rich in the boundary case, where delocalization proceeds through different renormalization patterns sharing the feature that the boundary field becomes irrelevant in the delocalized regime. The boundary delocalization (wetting) transition is shown to be continuous, with surface specific heat and layer thickness exponents which can take values that we determine.
We study the static and dynamic behavior of the one dimensional pair contact process with diffusion. Several critical exponents are found to vary with the diffusion rate, while the order-parameter moment ratio m=rho(2);/rho;(2) grows logarithmically with system size. The anomalous behavior of m is traced to a violation of scaling in the order parameter probability density, which in turn reflects the presence of two distinct sectors, one purely diffusive, the other reactive, within the active phase. Studies restricted to the reactive sector yield precise estimates for exponents beta and nu( perpendicular ), and confirm finite size scaling. We also determine the value m(c)=1.334 for the parity-conserving universality class in one dimension.