High-precision Estimate of the Critical Exponents for the Directed Ising Universality Class

Abstract

With extensive Monte Carlo simulations, we present high-precision estimates of the critical exponents of branching annihilating random walks with two offspring, a prototypical model of the directed Ising universality class in one dimension. To estimate the exponents accurately, we propose a systematic method to find corrections to scaling whose leading behavior is supposed to take the form $t^{-\chi }$ in the long-time limit at the critical point. Our study shows that $\chi \approx 0.75$ for the number of particles in defect simulations and $\chi \approx 0.5$ for other measured quantities, which should be compared with the widely used value of $\chi =1$. Using $\chi$ so obtained, we analyze the effective exponents to find that $\beta /{\nu }_{\Vert }=0.2872(2)$, $z=1.7415(5)$, $\eta =0.0000(2)$, and accordingly, $\beta /{\nu }_{\perp }=0.5000(6)$. Our numerical results for $\beta /{\nu }_{\Vert }$ and $z$ are clearly different from the conjectured rational numbers $\beta /{\nu }_{\Vert }=\frac{2}{7}\approx 0.2857$, $z=\frac{7}{4}=1.75$ by Jensen [Phys. Rev. E, {\bf 50}, 3623 (1994)]. Our result for $\beta /{\nu }_{\perp }$, however, is consistent with $\frac{1}{2}$, which is believed to be exact.

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