We investigate the dynamics of a two-dimensional axial next-nearest-neighbor Ising model following a quench to zero temperature. The Hamiltonian is given by H= -J_(0) summation operator(L)_(i,j=1)S_(i,j)S_(i+1,j)-J_(1)summation operator_(i,j=1)(S_{i,j}S_{i,j+1}-kappaS_{i,j}S_{i,j+2}) . For kappa<1 , the system does not reach the equilibrium ground state but slowly evolves to a metastable state. For kappa>1 , the system shows a behavior similar to that of the two-dimensional ferromagnetic Ising model in the sense that it freezes to a striped state with a finite probability. The persistence probability shows algebraic decay here with an exponent theta=0.235+/-0.001 while the dynamical exponent of growth z=2.08+/-0.01 . For kappa=1 , the system belongs to a completely different dynamical class; it always evolves to the true ground state with the persistence and dynamical exponent having unique values. Much of the dynamical phenomena can be understood by studying the dynamics and distribution of the number of domain walls. We also compare the dynamical behavior to that of a Ising model in which both the nearest and next-nearest-neighbor interactions are ferromagnetic.
We investigate the dynamics of a two-dimensional biaxial next-nearest-neighbor Ising model following a quench to zero temperature. The Hamiltonian is given by H=-J_{0}∑_{i,j=1}^{L}[(S_{i,j}S_{i+1,j}+S_{i,j}S_{i,j+1})-κ(S_{i,j}S_{i+2,j}+S_{i,j}S_{i,j+2})]. For κ<1, the system does not reach the equilibrium ground state and keeps evolving in active states forever. For κ≥1, though, the system reaches a final state, but it does not reach the ground state always and freezes to a striped state with a finite probability like a two-dimensional ferromagnetic Ising model and an axial next-nearest-neighbor Ising (ANNNI) model. The overall dynamical behavior for κ>1 and κ=1 is quite different. The residual energy decays in a power law for both κ>1 and κ=1 from which the dynamical exponent z has been estimated. The persistence probability shows algebraic decay for κ>1 with an exponent θ=0.22±0.002 while the dynamical exponent for ordering z=2.33±0.01. For κ=1, the system belongs to a completely different dynamical class with θ=0.332±0.002 and z=2.47±0.04. We have computed the freezing probability for different values of κ. We have also studied the decay of autocorrelation function with time for a different regime of κ values. The results have been compared with those of the two-dimensional ANNNI model.
We present a kinetic lattice system that evolves in general towards steady nonequilibrium states due to dynamical conflict between nearest- and next-nearest-neighbour interactions. Under two simple particular limits, the system would reach asymptotically the canonical equilibrium states for the ordinary Ising model and for the axial next-nearest-neighbour Ising (ANNNI) model, respectively. We find more generally that, independently of the lattice dimension, the steady state probability distribution for a given class of transition rates has a quasi-canonical structure with a short-range effective Hamiltonian. We solve exactly the one-dimensional version of this case, and compare its behaviour to the one for the ordinary ANNNI model. In particular, the system is shown to exhibit several spatially modulated phases and impure critical points. We also obtain some information on the phase diagram for the two-dimensional lattice.
The spin structure of an axial next-nearest-neighbor Ising (ANNNI) model in two dimensions (2D) is a renewed problem because different Monte Carlo (MC) simulation methods predicted different spin orderings. The usual equilibrium simulation predicts the occurrence of a floating incommensurate (IC) Kosterlitz-Thouless (KT) type phase, which never emerges in non-equilibrium relaxation (NER) simulations. In this paper, we first examine previously published results of both methods, and then investigate a higher transition temperature
Tc1 between the IC and paramagnetic phases. In the usual equilibrium simulation, we calculate the chain magnetization on larger lattices (up to $512\ifmmode\times\else\texttimes\fi{}512$ sites) and estimate ${T}_{c1}\ensuremath{\approx}1.16J$ with frustration ratio $\ensuremath{\kappa}(\ensuremath{\equiv}\ensuremath{-}{J}_{2}/{J}_{1})=0.6$. We examine the nature of the phase transition in terms of the Binder ratio
gL of spin overlap functions and the correlation-length ratio $\ensuremath{\xi}/L$. In the NER simulation, we observe the spin dynamics in equilibrium states by means of an autocorrelation function and also observe the chain magnetization relaxations from the ground and disordered states. These quantities exhibit an algebraic decay at $T\ensuremath{\lesssim}1.17J$. We conclude that the two-dimensional ANNNI model actually admits an IC phase transition of the KT type.
In this work, we study the critical behavior of second-order points, specifically the Lifshitz point (LP) of a three-dimensional Ising model with axial competing interactions [the axial-next-nearest-neighbor Ising (ANNNI) model], using time-dependent Monte Carlo simulations. We use a recently developed technique that helps us localize the critical temperature corresponding to the best power law for magnetization decay over time: ${\ensuremath{\langle}M\ensuremath{\rangle}}_{{m}_{0}=1}\ensuremath{\sim}{t}^{\ensuremath{-}\ensuremath{\beta}/\ensuremath{\nu}z}$, which is expected of simulations starting from initially ordered states. We obtain original results for the dynamic critical exponent
z, evaluated from the behavior of the ratio ${F}_{2}(t)={\ensuremath{\langle}{M}^{2}\ensuremath{\rangle}}_{{m}_{0}=0}/{\ensuremath{\langle}M\ensuremath{\rangle}}_{{m}_{0}=1}^{2}\ensuremath{\sim}{t}^{3/z}$, along the critical line up to the LP. We explore all the critical exponents of the LP in detail, including the dynamic critical exponent $\ensuremath{\theta}$ that characterizes the initial slip of magnetization and the global persistence exponent ${\ensuremath{\theta}}_{g}$ associated with 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 ${\ensuremath{\theta}}_{g}=0.336(4)$, values that are very different from those of the three-dimensional Ising model (the ANNNI model without the next-nearest-neighbor interactions at the
z axis, i.e.,
J2=0), i.e., $z\ensuremath{\approx}2.07$ and ${\ensuremath{\theta}}_{g}\ensuremath{\approx}0.38$. We also present estimates for the static critical exponents $\ensuremath{\beta}$ and $\ensuremath{\nu}$, obtained from extended time-dependent scaling relations. Our results for static exponents are in good agreement with previous works.