Universality-class crossover by a nonorder field introduced to the pair contact process with diffusion

The one-dimensional pair contact process with diffusion (PCPD), an interacting particle system with diffusion, pair annihilation, and creation by pairs, has defied consensus about the universality class to which it belongs. An argument by Hinrichsen [Physica A 361, 457 (2006)] claims that freely diffusing particles in the PCPD should play the same role as frozen particles when it comes to the critical behavior. Therefore, the PCPD is claimed to have the same critical phenomena as a model with infinitely many absorbing states that belongs to the directed percolation (DP) universality class. To investigate if diffusing particles are really indistinguishable from frozen particles in the sense of the renormalization group, we study numerically a variation of the PCPD by introducing a nonorder field associated with infinitely many absorbing states. We find that a crossover from the PCPD to DP occurs due to the nonorder field. By studying a similar model, we exclude the possibility that the mere introduction of a nonorder field to one model can entail a nontrivial crossover to another model in the same universality class, thus we attribute the observed crossover to the difference of the universality class of the PCPD from the DP class.

Figure 1

Semilogarithmic plots of ${\rho }_p\left(t\right)t^{\delta }$ vs $t$ around the critical point for $w={10}^{-4}$, where $\delta =0.1595$ is the critical decay exponent of the DP. For $p=0.13305$ (middle curve), the curve becomes flat from $t={10}^5$ for about three log decades and the other curves veer up ($p=0.133045$) or down ($p=0.133055$). Thus, we conclude that the critical point is $p_c=0.133050\left(5\right)$ and the model belongs to the DP class.

Figure 2

Double-logarithmic plots of ${\rho }_p$ vs $t$ at the critical points for various $w$'s from $w=0$ to 1 (bottom to top). Three curves corresponding to $w=0.5$, 0.9, and 1 are barely discernible, whereas differences among curves corresponding to $w<0.1$ are conspicuous.

Figure 3

Phase boundary of the PNF in the $wp$ plane. Symbols represent the critical points obtained by simulations (see Table 1), and lines that connect two nearest symbols are guides for the eyes. The phase boundary approaches the ordinate continuously with infinite slope, while it approaches the PCP critical point (corresponding to $w=1$) with finite slope. Inset: plot of $p_0-p_c$ vs $w$ on a double logarithmic scale. For comparison, Eq. (5) with $1/\varphi \approx 0.62$ is also drawn.

Figure 4

Plot of ${\rho }_b$ vs $t^{-\delta }$ at the critical point $p_c\left(w\right)$ for $w={10}^{-4}$. Close to the ordinate, the curve becomes almost straight. By a linear extrapolation using Eq. (6), we found the natural density ${\rho }_B$. The fitting result is drawn as a straight line. Inset: plot of ${\rho }_B$ vs $w$ on a double logarithmic scale. ${\rho }_B$ for small $w$ is fitted by a power function $a{w}^{\chi }$ with a result $\chi \approx 0.51$.

Figure 5

Double-logarithmic plots of ${\rho }_p$ vs $t$ at the critical point of the MPCP for various $\varepsilon$'s from $\varepsilon =0$ to 1 (top to bottom). As a guide to the eyes, a line segment with slope $-0.1595$ is also drawn. Unlike the PNF, the critical behavior for small $\varepsilon$ is hardly discernible from that for $\varepsilon =0$; see also Fig. 2. Inset: plot of $p_c$ as a function of $\varepsilon$ for the MPCP. Unlike the PNF, the slope at $\varepsilon =0$ is finite.