Block renormalization study on the nonequilibrium chiral Ising model

We present a numerical study on the ordering dynamics of a one-dimensional nonequilibrium Ising spin system with chirality. This system is characterized by a direction-dependent spin update rule. Pairs of $+-$ spins can flip to $++$ or $--$ with probability $(1-u)$ or to $-+$ with probability $u$ while $-+$ pairs are frozen. The system was found to evolve into the ferromagnetic ordered state at any $u<1$ exhibiting the power-law scaling of the characteristic length scale $\xi \sim t^{1/z}$ and the domain-wall density $\rho \sim t^{-\delta }$. The scaling exponents $z$ and $\delta$ were found to vary continuously with the parameter $u$. To establish the anomalous power-law scaling firmly, we perform the block renormalization analysis proposed by Basu and Hinrichsen [U. Basu and H. Hinrichsen, J. Stat. Mech.: Theor. Exp. (2011) P11023]. The block renormalization method predicts, under the assumption of dynamic scale invariance, a scaling relation that can be used to estimate the scaling exponent numerically. We find the condition under which the scaling relation is justified. We then apply the method to our model and obtain the critical exponent $z\delta$ at several values of $u$. The numerical result is in perfect agreement with that of the previous study. This study serves as additional evidence for the claim that the nonequilibrium chiral Ising model displays power-law scaling behavior with continuously varying exponents.

Figure 1

Space-time patterns of the Ising spins with chirality ($\overline{u}=\overline{v}=0$ and $u=1-v=0.5$) in (a) and (b), and without chirality $\left(u=v=\overline{u}=\overline{v}=0.5\right)$ in (c) and (d). Spin dynamics are shown in (a) and (c), where black (white) pixels represent sites of $+(-)$ states. Particle dynamics are shown in (b) and (d), where orange (light) and blue (dark) pixels represent $A$ and $B$ particles, respectively. The horizontal and vertical directions correspond to the spatial and temporal directions, respectively.

Figure 2

Temporal decay of the two-block correlation function $P_c(b=4)$ at the model parameter $u=0.3$. Also shown is the overall particle density $\rho \left(t\right)$.

Figure 3

(a) $S_{OB}(b,t)$ for $b=1,2,...,32$ (from top to bottom) and (b) $S_{OB}\left(b\right)$ at the model parameter $u=0.3$.

Figure 4

$S_c$ for two-blocks patterns $c$. Lines are a guide to the eye.

Figure 5

Correlation functions $C(r,t)$ (top three curves) and $F(r,t)$ (middle three curves), and their product (bottom three curves). The dot-dashed lines with indicated slopes are guides to the eyes. The data are obtained for the NCIM with the parameter values $u=1-v=0.3$ and $\overline{u}=\overline{v}=0$. The lattice size is $L=2^{20}$ and the simulation time is up to $t=2^{25}$. The data are averaged over more than 100 samples and log-binned.

Figure 6

Comparison of the scaling exponent $z\delta$ obtained from the Monte Carlo simulation of Ref. [5] and the block renormalization analysis.