Ising spin model

There is a fundamental relationship between d-dimensional PCA and d + 1)-dimensional Ising spin models. The simplest way to make the connection is to think of the successive temporal layers of the PCA as successive hyper-planes of the next higher-dimensional spatial lattice. Because the PCA rules (at least the set of PCA rules that we will be dealing with) are (1) Markovian (i.e. the probability of a state at time t + T depends only on a set of states at time t, and (2) local, one can always define a Hamiltonian on the higher-dimensioned spatial lattice such that the thermodynamic weight of a configuration 5j,( is equal to the probability of a corresponding space-time history Si t). ... 

In 1987, Swendsen and Wang (SW) [3] introduced a new Monte Carlo algorithm for the Ising spin model, which constituted a radical departure from the Metropolis or single-spin flip method used until then. Since the recipe is relatively straightforward, it is instructive to begin with a description of this algorithm. 

The Ising spin model does not consider the coexistence of the ordered (ferromagnetic) and non-ordered (paramagnetic) phases at subcritical temperatures. As a result, there is no latent heat r/T) and disorder parameter associated with the ferromagnetic transition. The condition dh/dT = 0 must be added to h=0. The known CXC-dependence of the lattice-gas chemical potential ... 

Fig. 69. (a) Part of the body-centered cubic lattice ordered in the B2 structure (left part) and in the Dtp structure (right part). Left part shows assignment of four sublattices a, b, c and d, In the B2 structure (cf. also fig. 66a), the concentrations of A atoms are the same at the a and c sublatticcs, but differ from the concentrations of the b, d sublattices, while in the DOj structure the concentration of the b sublattice differs from that of the d sublatlice, but both differ from those of the a, c sublattices (which are still the same). In terms of an Ising spin model, these sublattice concentrations translate into sublattice magnetizations mu, mu, mc, m,i, which allow to define three order parameter components / = ma + mL- — mu — m,/, fa = m - mc + mu — m,j, and fa = -ma + m., + mu — nij. 

More recently suggested models for bulk systems treat oil, water and amphiphiles on equal footing and place them all on lattice sites. They are thus basically lattice models for ternary fluids, which are generalized to capture the essential properties of the amphiphiles. Oil, water, and amphiphiles are represented by Ising spins 5 = -1,0 and +1. If one considers all possible nearest-neighbor interactions between these three types of particle, one obtains a total number of three independent interaction parameters, and... 

In both of these cases, represents the energy of two noninteracting Ising spin systems, one system having spins s over one sublattice and the other spins s over the other sublattice. Indeed, the Q2R CA actually provides an efficient microcanonical algorithm for performing parallel simulation of the Ising model (see discussion in section 7.1.5). 

It was shown by Wilson [131] that the Kadanoff procedure, combined with the Landau model, may be used to identify the critical point, verify the scaling law and determine the critical exponents without obtaining an exact solution, or specifying the nature of fluctuations near the critical point. The Hamiltonian for a set of Ising spins is written in suitable units, as before... 

This nanoparticle sample exhibits strong anisotropy, due to the uniaxial anisotropy of the individual particles and the anisotropic dipolar interaction. The relative timescales (f/xm) of the experiments on nanoparticle systems are shorter than for conventional spin glasses, due to the larger microscopic flip time. The nonequilibrium phenomena observed here are indeed rather similar to those observed in numerical simulations on the Ising EA model [125,126], which are made on much shorter time (length) scales than experiments on ordinary spin glasses [127]. 

According to RG theory [11, 19, 20], universality rests on the spatial dimensionality D of the systems, the dimensionality n of the order parameter (here n = 1), and the short-range nature of the interaction potential 0(r). In D = 3, short-range means that 0(r) decays as r p with p>D + 2 — tj = 4.97 [21], where rj = 0.033 is the exponent of the correlation function g(r) of the critical fluctuations [22] (cf. Table I). Then, the critical exponents map onto those of the Ising spin-1/2 model, which are known from RG calculations [23], series expansions [11, 12, 24] and simulations [25, 26]. For insulating fluids with a leading term of liquid metals [27-29] the experimental verification of Ising-like criticality is unquestionable. 

Fig. 7. Mean-field phase diagram of the four-state Ising-Potts model (A/kB = 90 K, J lkB = 125 K). The high-spin (HS) phase, the low-spin (LS) phase, and the ferroelectric-ordered (FO) phase are shown. The arrow line corresponds to Jo/kB = — 36 K, appropriate to the [Mn(taa)] system.

Kawamoto T, Abe S. Thermal hysteresis loop of the spin-state in nanoparticles of transition metal complexes Monte Carlo simulations on an Ising-like model. Chemical Communications. 2005 No. 31, 3933-3935. DOI 10.1039/b506643c. 

Atitoaie A, Tanasa R, Enachescu C. Size dependent thermal hysteresis in spin crossover nanoparticles reflected within a Monte-Carlo based Ising-like model. Journal of Magnetism and Magnetic Materials. 2012 324 1596-1600. DOI 10.1016/j.jmmm. 2011.12.011. 

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