Contact process on generalized Fibonacci chains: Infinite-modulation criticality and double-log periodic oscillations

We study the nonequilibrium phase transition of the contact process with aperiodic transition rates using a real-space renormalization group as well as Monte Carlo simulations. The transition rates are modulated according to the generalized Fibonacci sequences defined by the inflation rules A $\to$ AB${}^k$ and B $\to$ A. For $k=1$ and 2, the aperiodic fluctuations are irrelevant, and the nonequilibrium transition is in the clean directed percolation universality class. For $k\ge 3$, the aperiodic fluctuations are relevant. We develop a complete theory of the resulting unconventional “infinite-modulation” critical point, which is characterized by activated dynamical scaling. Moreover, observables such as the survival probability and the size of the active cloud display pronounced double-log periodic oscillations in time which reflect the discrete scale invariance of the aperiodic chains. We illustrate our theory by extensive numerical results, and we discuss relations to phase transitions in other quasiperiodic systems.

Figure 1

Sequence of transition rates for the contact process on a generalized Fibonacci chain, showing the fourth generation of the $k=2$ chain.

Figure 2

Survival probability $P_s$ and size $N_s$ of the active cloud vs. time $t$ for $k=1$ and strong inhomogeneity ${\lambda }_A/{\lambda }_B=0.01$. The data are averages of 100 000 (away from criticality) to 500 000 (at criticality) trials. The critical point is located at ${\lambda }_B=57.97$. The solid straight lines represent power-law fits giving the critical exponents $\delta =0.160$ and $\Theta =0.314$.

Figure 3

Survival probability $P_s$ and size $N_s$ of the active cloud vs. time $t$ for the case $k=2$ and weak inhomogeneity ${\lambda }_A/{\lambda }_B=2/3$. The data are averages of 100 000 to 150 000 trials. The critical point is located at ${\lambda }_B=4.0408$. The solid straight lines represent power-law fits giving the critical exponents $\delta =0.158$ and $\Theta =0.311$.

Figure 4

$N_s$ versus $P_s$ for the case $k=2$ and two different inhomogeneity strengths, ${\lambda }_A/{\lambda }_B=2/3$ (upper panel) and ${\lambda }_A/{\lambda }_B=0.01$ (lower panel). The data are averages of 100 000 to 150 000 trials. The solid line in the upper panel is a power-law fit of the critical curve (${\lambda }_B=4.0408$) yielding $\Theta /\delta =1.971$. The dashed line in the lower panel represents a power law with the clean exponent $-{\Theta }_{\mathrm{DP}}/{\delta }_{\mathrm{DP}}=-1.96712$.

Figure 5

Example of a density decay run, starting from a fully active lattice of 15 generations of the $k=3$ chain (173 383 sites). The inhomogeneity strength is ${\lambda }_A/{\lambda }_B=0.04$. In the main panel, dark blue dots denote active sites while light yellow marks inactive sites. The left panel shows the corresponding density $\rho$ of active sites. The horizontal lines are located at times that correspond to the inverse transition rates at different renormalization group steps, $t={\lambda }_B^{-1},{\mu }^{-1}$.

Figure 6

Survival probability $P_s$ and size $N_s$ of the active cloud vs. time $t$ for the case $k=3$ and ${\lambda }_A/{\lambda }_B=0.04$ (5000 trials). The steps and plateaus in the critical curve, ${\lambda }_B=13.12$, become more pronounced with increasing time. They can be associated with the discrete values of $\lambda$ and $\mu$ appearing in the renormalization group (marked by large stars and hexagons).

Figure 7

Upper panel: $N_s$ versus $P_s$ for the case $k=3$ and ${\lambda }_A/{\lambda }_B=0.04$ (5000 trials). The maximum time is $t_{\max }=1.4\times {10}^9$ at criticality. The critical curve, ${\lambda }_B=13.12$, shows pronounced steps as predicted in Sec. 3e. Lower panel: Critical curves for several inhomogeneity strengths ${\lambda }_A/{\lambda }_B$ (5000 to 100 000 trials).

Figure 8

Phase diagram of the contact process for $k=3$. The dots are the Monte Carlo results for ${\lambda }_A/{\lambda }_B=0.001$, 0.004, 0.01, 0.04, 0.1, and 1. The solid line represents the renormalization group result Eq. (23).

Figure 9

Quantitative analysis of the critical $N_s$ versus $P_s$ curve for $k=3$ and ${\lambda }_A/{\lambda }_B=0.04$ (maximum time $t_{\max }=1.4\times {10}^9$) and ${\lambda }_A/{\lambda }_B=25$ (maximum time $t_{\max }=2\times {10}^8$). The solid line is a fit of the envelope of the curve to the power law $N_s\sim P_s^{-\overline{\Theta }/\overline{\delta }}$ yielding $\overline{\Theta }/\overline{\delta }=3.79$.