Branching annihilating random walk with long-range repulsion: logarithmic scaling, reentrant phase transitions, and crossover behaviors

Abstract

We study absorbing phase transitions in the one-dimensional branching annihilating random walk with long-range repulsion. The repulsion is implemented as hopping bias in such a way that a particle is more likely to hop away from its closest particle. The bias strength due to long-range interaction has the form $\varepsilon x^{-\sigma },$ where x is the distance from a particle to its closest particle, $0\le \sigma \le 1,$ and the sign of $\varepsilon$ determines whether the interaction is repulsive (positive $\varepsilon )$ or attractive (negative $\varepsilon )$. A state without particles is the absorbing state. We find a threshold $\varepsilon _s,$ such that the absorbing state is dynamically stable for small branching rate q if $\varepsilon < \varepsilon _s.$ The threshold differs significantly, depending on parity of the number $\ell$ of offspring. When $\varepsilon >\varepsilon _s,$ the system with odd $\ell$ can exhibit reentrant phase transitions from the active phase with nonzero steady-state density to the absorbing phase, and back to the active phase. On the other hand, the system with even $\ell$ is in the active phase for nonzero q if $\varepsilon >\varepsilon _s.$ Still, there are reentrant phase transitions for $\ell =2.$ Unlike the case of odd $\ell ,$ however, the reentrant phase transitions can occur only for $\sigma =1$ and $0<\varepsilon < \varepsilon _s.$ We also study the crossover behavior for $\ell = 2$ when the interaction is attractive (negative $\varepsilon )$, to find the crossover exponent $\phi =1.123(13)$ for $\sigma =0.$