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 epsilon x(-sigma), where x is the distance from a particle to its closest particle, 0 <= sigma <= 1, and the sign of epsilon determines whether the interaction is repulsive (positive epsilon) or attractive (negative epsilon). A state without particles is the absorbing state. We find a threshold epsilon(s), such that the absorbing state is dynamically stable for small branching rate q if epsilon < epsilon(s). The threshold differs significantly, depending on parity of the number l of offspring. When epsilon > epsilon(s), the system with odd l 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 l is in the active phase for nonzero q if epsilon > epsilon(s). Still, there are reentrant phase transitions for l = 2. Unlike the case of odd l, however, the reentrant phase transitions can occur only for sigma = 1 and 0 < epsilon < epsilon(s). We also study the crossover behavior for l = 2 when the interaction is attractive (negative epsilon), to find the crossover exponent phi = 1.123(13) for sigma = 0.