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

Abstract: 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