The study of the phase ordering kinetics of the ferromagnetic one-dimensional Ising model dates back to 1963 for non conserved order parameter (NCOP) and to 1991 for conserved order parameter (COP). The case of long range interactions $J(r)$ has been widely studied at equilibrium but their effect on relaxation is a much less investigated field. Here we make a detailed numerical and analytical study of both cases, NCOP and COP. Many results are valid for any positive, decreasing coupling $J(r)$, but we focus specifically on the exponential case, $J_{exp}(r)=e^{-r/R}$ with varying $R>0$, and on the integrable power law case, $J_{pow}(r) =1/r^{1+σ}$ with $σ> 0$. We find that the {\it asymptotic} growth law $L(t)$ is the usual algebraic one, $L(t)\sim t^{1/z}$, of the corresponding model with nearest neighborg interaction ($z_{NCOP}=2$ and $z_{COP}=3$) for all models except $J_{pow}$ for small $σ$: in the non conserved case when $σ\le 1$ ($z_{NCOP}=σ+1$) and in the conserved case when $σ\to 0^+$ ($z_{COP}=4β+3$, where $β=1/T$ is the inverse of the absolute temperature). The models with space decaying interactions also differ markedly from the ones with nearest neighbors due to the presence of many long-lasting preasymptotic regimes, such as an exponential mean-field behavior with $L(t)\sim e^{t}$, a ballistic one with $L(t)\sim t$, a slow (logarithmic) behavior $L(t)\sim \ln t$ and one with $L(t)\sim t^{1/σ+1}$. All these regimes and their validity ranges have been found analytically and verified in numerical simulations. Our results show that the main effect of the conservation law is a strong slowdown of COP dynamics if interactions have an extended range. Finally, we compare the Ising model at hand with continuum approaches based on a Ginzburg-Landau free energy.
We study two-dimensional Ising spins, evolving through reinforcement learning using their state, action, and reward. The state of a spin is defined as whether it is in the majority or minority with its nearest neighbours. The spin updates its state using an ε-greedy algorithm. The parameter ε plays the role equivalent to the temperature in the Ising model. We find a phase transition from long-ranged ordered to a disordered state as we tune ε from small to large values. In analogy with the phase transition in the Ising model, we calculate the critical ε and the three critical exponents β, γ, ν …
The frustration properties of the Ising model on a one-dimensional monoatomic equidistant lattice are investigated taking into account the exchange interactions of atomic spins at the sites of the first (nearest), second (next-nearest) and third neighbors. The exact solution of the model was obtained using the Kramers--Wannier transfer matrix method. The types of magnetic ordering of the ground state of the model are determined, and a magnetic phase diagram is constructed. Criteria are formulated for the occurrence of magnetic frustrations in the presence of competition between the energies of…
The present work clarifies a failure of the effective-field theory in predicting a false spontaneous long-range order and phase transition of Ising nanoparticles, nanoislands, nanotubes and nanowires with either zero- or one-dimensional magnetic dimensionality. It is conjectured that the standard formulation of the effective-field theory due to Honmura and Kaneyoshi generally predicts for the Ising spin systems a spurious spontaneous long-range order with nonzero critical temperature regardless of their magnetic dimensionality whenever at least one Ising spin has coordination number greater th…
In this paper, the frustration properties of the Ising model on a one-dimensional monoatomic equidistant lattice in an external magnetic field are investigated, taking into account the exchange interactions of atomic spins at the sites of the first, second, and third neighbors. Exact analytical expressions for the thermodynamic functions of the system are obtained by the Kramers--Wannier transfer-matrix method. A magnetic phase diagram of the ground state of such a spin system is constructed and studied thoroughly. The points and lines of frustrations of the system depending on the values and …
In a ferromagnetic Ising system, domain pattern formation, i.e., phase-ordering, occurs after a sudden quench. We propose the method to simulate the pattern formation dynamics by an Ising machine. We demonstrate that the method reproduces domain patterns similar to those simulated by the Monte Carlo method. Moreover, the same domain growth law is observed in the proposed method and the Monte Carlo method.
We study the possibility of long-range ordering (LRO) in a 3D system of vertically stacked layers of Ising antiferromagnet on a kagome lattice (SIAKL). Monte Carlo simulations are carried out for a varying interlayer coupling strength and a finite-size scaling analysis is performed for selected cases. Unlike in the related Ising system on a triangular lattice, in which even a finite number of the layers can stabilize LRO, or the Heisenberg system on the same kagome lattice, in which LRO emerged in 3D for a sufficient strength of the interlayer coupling, no LRO could be observed in the present …
