A brief account of the Ising and Ising-like models: Mean-field, effective-field and exact results

Strecka, Jozef; Jascur, Michal

doi:10.48550/arxiv.1511.03031

Cited by 10 publications

(19 citation statements)

References 276 publications

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“…It is important to mention that figure 3 presents some resemblances with results from references [24,36,40] concerning the second-order transition lines. Also, by increasing the value of the coordination number z, the ferrimagnetic domain F becomes important.…”

Section: Finite-temperature Phase Diagramssupporting

confidence: 67%

“…Few exactly solved mixed-Ising models exist in the literature. For recent review on the subject, the reader should refer to references [32][33][34][35][36][37][38][39][40][41]. Experimentally, the investigation of such systems has been performed for many years and has shown strong effects of the external constraint on the physical properties of the system [42,43].…”

Section: Introductionmentioning

confidence: 99%

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Bethe lattice approach study of the mixed spin-1/2 and spin-7/2 Ising model in a longitudinal magnetic field

Eddahri¹,

Karimou²,

Razouk³

et al. 2018

Condens. Matter Phys.

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The magnetic properties of the mixed spin-1 2 and spin-7 2 Ising model with a crystal-field in a longitudinal magnetic field are investigated on the Bethe lattice using exact recursion relations. The ground-state phase diagram is constructed. The temperature-dependent one is displayed in the case of uniform crystal-field on the (k B T/|J |, D/|J |) plane in the absence of the external constraint for lattice coordination numbers z = 3, 4, 6.The order parameters and corresponding response functions as well as the internal energy are calculated and examined in detail in order to feature the real nature of phase boundaries and corresponding temperatures. The thermal variations of the average magnetization are classified according to the Néel nomenclature.

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Section: Finite-temperature Phase Diagramssupporting

confidence: 67%

Section: Introductionmentioning

confidence: 99%

Bethe lattice approach study of the mixed spin-1/2 and spin-7/2 Ising model in a longitudinal magnetic field

Eddahri¹,

Karimou²,

Razouk³

et al. 2018

Condens. Matter Phys.

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“…Both Monte-Carlo methods [2] and mean-field methods [3] have been employed to study this problem. Another example of the application of such models is the contact process which is a stochastic process describing the growth of a population on a structured or unstructured lattice.…”

Section: Introductionmentioning

confidence: 99%

Data-Driven Optimal Closures for Mean-Cluster Models: Beyond the Classical Pair Approximation

Ahmadi,

Foster,

Protas

2022

Preprint

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This study concerns the mean-clustering approach to modelling the evolution of lattice dynamics.Instead of tracking the state of individual lattice sites, this approach describes the time evolution of the concentrations of different cluster types. It leads to an infinite hierarchy of ordinary differential equations which must be closed by truncation using a so-called closure condition. This condition approximates the concentrations of higher-order clusters in terms of the concentrations of lowerorder ones. The pair approximation is the most common form of closure. Here, we consider its generalization, termed the "optimal approximation", which we calibrate using a robust data-driven strategy. To fix attention, we focus on a recently proposed structured lattice model for a nickelbased oxide, similar to that used as cathode material in modern commercial Li-ion batteries. The form of the obtained optimal approximation allows us to deduce a simple sparse closure model. In addition to being more accurate than the classical pair approximation, this "sparse approximation" is also physically interpretable which allows us to a posteriori refine the hypotheses underlying construction of this class of closure models. Moreover, the mean-cluster model closed with this sparse approximation is linear and hence analytically solvable such that its parametrization is straightforward. On the other hand, parametrization of the mean-cluster model closed with the pair approximation is shown to lead to an ill-posed inverse problem.

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“…As we have mentioned earlier, the phase structure remains the same for small values of ω in the BEG mdoel. Referring the readers to [2,8] for detailed derivations, we briefly outline here some mathematical relations pertaining to the phase structure. Firstly, in the Blume-Capel limit the zero field expression for magnetization simplifies considerably, m = 2 sinh(βm) 2 cosh(βm) + e β D (Blume-Capel, H = 0) (10) and in the vicinity of the critical point the Landau expansion of the free energy gives…”

Section: Introductionmentioning

confidence: 99%

Thermodynamic geometry of the spin-1 model. II. Criticality and coexistence in the mean field approximation

Sahay¹,

Sanwari²

2021

Preprint

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We continue our study of the thermodynamic geometry of the spin one model from [1] (paper I) by probing the state space geometry of the Blume Emery Griffiths (BEG) model, and its limiting case of the Blume Capel model, in their mean field approximation. By accounting for the stochastic variables involved we construct from the thermodynamic state space two complimentary two-dimensional geometries with curvatures R m and R q which are shown to encode correlations in the model's two order parameters, namely, the magnetization m and the quadrupole moment q. The geometry is investigated in the zero as well as the non zero magnetic field region. We find that the relevant scalar curvatures diverge to negative infinity along the critical lines with the correct scaling and amplitude. We then probe the geometry of phase coexistence and find that the relevant curvatures predict the coexistence curve remarkably well via their respective R-crossing diagrams. We also briefly comment on the effectiveness of the geometric correlation length compared to the commonly used Ornstein-Zernicke type correlation length vis-a-vis their scaling properties.

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A brief account of the Ising and Ising-like models: Mean-field, effective-field and exact results

Cited by 10 publications

References 276 publications

Bethe lattice approach study of the mixed spin-1/2 and spin-7/2 Ising model in a longitudinal magnetic field

Bethe lattice approach study of the mixed spin-1/2 and spin-7/2 Ising model in a longitudinal magnetic field

Data-Driven Optimal Closures for Mean-Cluster Models: Beyond the Classical Pair Approximation

Thermodynamic geometry of the spin-1 model. II. Criticality and coexistence in the mean field approximation

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