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

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 $\sigma=0$.