Short-Time Critical Dynamics of Damage Spreading in the Two-Dimensional Ising Model

TL;DR

This study investigates the short-time dynamics of damage spreading in the 2D Ising model at criticality using Monte Carlo simulations, revealing distinct growth exponents and epidemic behaviors.

Contribution

It provides new insights into damage propagation exponents and epidemic spreading characteristics at the critical temperature and below in the 2D Ising model.

Findings

Damage increases with an exponent of approximately 1.915 at criticality.

Epidemic spread exhibits a growth exponent around 1.9 and a plateau in survival probability.

Critical damage spreading occurs at about 0.51 times the critical temperature with specific power-law exponents.

Abstract

The short-time critical dynamics of propagation of damage in the Ising ferromagnet in two dimensions is studied by means of Monte Carlo simulations. Starting with equilibrium configurations at $T= \infty$ and magnetization $M=0$, an initial damage is created by flipping a small amount of spins in one of the two replicas studied. In this way, the initial damage is proportional to the initial magnetization $M_0$ in one of the configurations upon quenching the system at $T_C$, the Onsager critical temperature of the ferromagnetic-paramagnetic transition. It is found that, at short times, the damage increases with an exponent $\theta_D=1.915(3)$, which is much larger than the exponent $\theta=0.197$ characteristic of the initial increase of the magnetization $M(t)$. Also, an epidemic study was performed. It is found that the average distance from the origin of the epidemic ($\langle R^2(t)\rangle$) grows with an exponent $z^* \approx \eta \approx 1.9$, which is the same, within error bars, as the exponent $\theta_D$. However, the survival probability of the epidemics reaches a plateau so that $\delta=0$. On the other hand, by quenching the system to lower temperatures one observes the critical spreading of the damage at $T_{D}\simeq 0.51 T_C$, where all the measured observables exhibit power laws with exponents $\theta_D = 1.026(3)$, $\delta = 0.133(1)$, and $z^*=1.74(3)$.