Time-dependent Monte Carlo simulations of the critical and Lifshitz points of the ANNNI model

In this work, we study the critical behavior of second order points and specifically of the Lifshitz point (LP) of a three-dimensional Ising model with axial competing interactions (ANNNI model), using time-dependent Monte Carlo simulations. First of all, we used a recently developed technique that helps us localize the critical temperature corresponding to the best power law for magnetization decay over time: $\left\langle M\right\rangle _{m_{0}=1}\sim t^{-\beta /\nu z}\ $ which is expected of simulations starting from initially ordered states. Secondly, we obtain original results for the dynamic critical exponent $z$, evaluated from the behavior of the ratio $F_{2}(t)=\left\langle M^{2}\right\rangle _{m_{0}=0}/\left\langle M\right\rangle _{m_{0}=1}^{2}\sim t^{3/z}$, along the critical line up to the LP. Finally, we explore all the critical exponents of the LP in detail, including the dynamic critical exponent $\theta $ that characterizes the initial slip of magnetization and the global persistence exponent $\theta _{g}$ associated to the probability $P(t)$ that magnetization keeps its signal up to time $t$. Our estimates for the dynamic critical exponents at the Lifshitz point are $z=2.34(2)$ and $\theta _{g}=0.336(4)$, values very different from the 3D Ising model (ANNNI model without the next-nearest-neighbor interactions at $z$-axis, i.e., $J_{2}=0$) $z\approx 2.07$ and $\theta _{g}\approx 0.38$. We also present estimates for the static critical exponents $\beta $ and $\nu $, obtained from extended time-dependent scaling relations. Our results for static exponents are in good agreement with previous works