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

TL;DR

This paper uses extensive Monte Carlo simulations and a systematic correction-to-scaling method to precisely estimate critical exponents of the directed Ising universality class, revealing deviations from previous conjectures.

Contribution

It introduces a systematic approach to determine correction-to-scaling exponents, leading to more accurate critical exponent estimates for the directed Ising class.

Findings

Estimated critical exponents differ from earlier conjectures.

Identified correction-to-scaling exponent $ ext{chi}$ around 0.75 and 0.5.

Confirmed some exponents are consistent with exact values.

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 $\beta/\nu_\| = 0.2872(2)$, $z = 1.7415(5)$, $\eta = 0.0000(2)$, and accordingly, $\beta /\nu_\perp = 0.5000(6)$. Our numerical results for $\beta/\nu_\|$ and $z$ are clearly different from the conjectured rational numbers $\beta/\nu_\| = \frac{2}{7} \approx 0.2857$, $z = \frac{7}{4}= 1.75$ 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.