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We study absorbing phase transitions in the one-dimensional branchingannihilating random walk with long-range repulsion. The repulsion isimplemented as hopping bias in such a way that a particle is more likely to hopaway from its closest particle. The bias strength due to long-range interactionhas the form $\varepsilon x^{-\sigma}$, where $x$ is the distance from aparticle 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 withoutparticles is the absorbing state. We find a threshold $\varepsilon_s$ such thatthe absorbing state is dynamically stable for small branching rate $q$ if$\varepsilon < \varepsilon_s$. The threshold differs significantly, dependingon parity of the number $\ell$ of offspring. When $\varepsilon>\varepsilon_s$,the system with odd $\ell$ can exhibit reentrant phase transitions from theactive phase with nonzero steady-state density to the absorbing phase, and backto the active phase. On the other hand, the system with even $\ell$ is in theactive phase for nonzero $q$ if $\varepsilon>\varepsilon_s$. Still, there arereentrant 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 thecrossover exponent $\phi=1.123(13)$ for $\sigma=0$.

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