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