Critical behavior of the Ising model with four-spin coupling

doi:10.1016/S0304-8853(97)90016-0

Journal of Magnetism and Magnetic Materials

Volume 88, Issues 1–2, 2 July 1990, Pages 87-92

https://doi.org/10.1016/S0304-8853(97)90016-0 Get rights and content

Abstract

Critical conditions for the first-order phase transition are determined for a system of $12$ spins on the Ising model with a four-spin coupling term. The critical point parameters <S^z>_c, T_c and E_c are found from the mean field theory. Relations between the three critical values and the four-spin coupling strength are discussed.

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Cited by (34)

Frustration in an exactly solvable mixed-spin Ising model with bilinear and three-site four-spin interactions on a decorated square lattice
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The investigation of multi-spin interactions has been initiated several decades ago in order to clarify their influence on phase transitions and magnetic properties in various physical systems. In order to investigate basic aspects of multi-spin interactions the authors have utilized various theoretical methods including exact calculations [1–15], series expansions [16–18], renormalization-group techniques [19–22], Monte Carlo simulations [23–26], mean-field and effective-field theory [27–30]. The main aim of this work is to extend our recent research in this field in order to investigate in detail the role of so-called three-site four-spin interactions in crystalline systems with localized magnetic moments.
Competitive effects of so-called three-site four-spin interactions, single ion anisotropy and bilinear interactions is studied in the mixed spin-1/2 and spin-1 Ising model on a decorated square lattice. Exploring the decoration–iteration transformation, we have obtained exact closed-form expressions for the partition function and other thermodynamic quantities of the model. From these relations, we have numerically determined ground-state and finite-temperature phase diagrams of the system. We have also investigated temperature variations of the correlation functions, internal energy, entropy, specific heat and Helmholtz free energy of the system. From the physical point of view, the most interesting result represents our observation of a partially ordered ferromagnetic or phase in the system with zero bilinear interactions. It is remarkable, that due to strong frustrations disordered spins survive in the system even at zero temperature, so that the ground state of the system becomes macroscopically degenerate with non-zero entropy. Introduction of arbitrarily small bilinear interaction completely removes degeneracy and the entropy always goes to zero at the ground state.
Monte Carlo investigation of the mixed spin Ising model with four-spin interaction and next-nearest neighbor couplings
2015, Superlattices and Microstructures
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For instance, the violation of universality is observed in the two-dimensional square lattice Ising model with nearest-neighbor, next-nearest neighbor, and four spin interactions [5,6]. From the theoretical point of view, the monoatomic Ising models with multispin interactions have been investigated in detail within different methods, such as mean field approximation [7,8], effective field theory [9,10], some more accurate treatments such as series expansion [11,12], renormalization group methods [13], Monte Carlo simulations [14], and also exact calculations [15,16]. Experimentally, an interesting fact for the models with multispin interactions has been reported.
A Monte Carlo method is used to investigate the phase diagram of the ferromagnetic two-dimensional mixed spin-1 and spin-1/2 Ising model with four-spin interaction J₄ and next-nearest couplings J′. In the absence of this latter interaction, the system does not present any evidence for the tricritical point, whereas such behavior was suggested by finite cluster approximation. In the presence of the next-nearest neighbor interaction, the phase diagram is qualitatively and quantitatively different from that obtained with J′ = 0. It undergoes two kinds of behaviors according to the negative and positive value of J′. In particular, for J₄/J₂ = −4 the system keeps the coexistence of the two ground states up to a J′-dependent finite temperature. In addition to the phase diagrams, the thermal dependence of the magnetizations have also been examined. Some characteristic features have been found which depend on the strengths of the interactions J₄ and J′.
Thermodynamical properties of the mixed spin transverse Ising model with four-spin interactions
2014, Physica A: Statistical Mechanics and its Applications
The thermodynamical properties of the mixed spin transverse Ising system consisting of spin- $1 / 2$ and spin-1 with four-spin interactions are studied within the frame work of the finite cluster approximation based on single-site cluster theory. In this approach, the state equations are derived for the two-dimensional square lattice. In addition to the phase diagrams which show qualitatively interesting features (variety of transitions and tricritical behaviour), we find some characteristic behaviours of the longitudinal and transverse sublattice magnetizations. In particular, the gap of the longitudinal magnetization at first order transition decreases with increasing values of the transverse field. Moreover, the effects of the transverse field on the internal energy and the specific heat are also investigated.
Magnetic properties of a ferromagnetic thin film with four spin interaction: A Monte Carlo simulation study
2013, Journal of Magnetism and Magnetic Materials
Monte Carlo simulation has been used to study the critical behaviors and the magnetic properties of a ferromagnetic thin Ising film with a plaquette four spin interaction. The effects of the ratio $r_{s} = J_{s} / J$ of the surface exchange interaction to the bulk one and the four spin interaction on phase diagrams are investigated. A number of characteristic behaviors have been found, which include the first- and second-order phase transitions, thus also the tricritical points, triple point and isolated critical point.
Thermodynamical properties of the mixed spin Blume-Capel model with four-spin interactions
2013, Journal of Magnetism and Magnetic Materials
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For instance, they display the nonuniversal critical phenomena [2,3]. From the theoretical point of view, monoatomic Ising models with multispin interactions have been studied within different methods, such as mean field approximation [4,5], effective field theory [6–8], series expansions [9,10], renormalization group methods [11], Monte Carlo simulations [12] and exact calculations [13]. Experimentally, the models with multipsin interactions can be used to describe different physical systems such as classical fluids [14], solid 3He [15], lipid bilayers [16], metamagnets [17] and rare gases [18].
A mixed spin Blume–Capel model consisting of spin-1/2 and spin-1 with four-spin interactions and crystal field interaction is studied by the use of the finite cluster approximation based on a single-site cluster theory. In this approach, the state equations are derived for the two dimensional square lattice. In addition to the phase diagrams and magnetizations which show a variety of interesting features, the influence of the four-spin interaction on the internal energy, specific heat and zero-field magnetic susceptibility is also examined.
Mixed spin Ising model with four-spin interaction and random crystal field
2012, Physica B: Condensed Matter
Citation Excerpt :
For instance, they display the nonuniversal critical phenomena [2,3], and deviations from the bloch T3/2 law at low temperatures [4]. Theoretically, monoatomic Ising models, with multispin interactions have been studied within different methods, such as mean field approximation [5,6], effective field theory [7–9], series expansions [10,11], renormalization group methods [12], Monte Carlo simulations [13], and exact calculations [14]. From the experimental point of view, the models with multispin interactions can be used to describe various physical systems such as classical fluids [15], solid 3He [16], lipid bilayers [17], and rare gases [18].
The effects of fluctuations of the crystal field on the phase diagram of the mixed spin-1/2 and spin-1 Ising model with four-spin interactions are investigated within the finite cluster approximation based on a single-site cluster theory. The state equations are derived for the two-dimensional square lattice. It has been found that the system exhibits a variety of interesting features resulting from the fluctuation of the crystal field interactions. In particular, for low mean value D of the crystal field, the critical temperature is not very sensitive to fluctuations and all transitions are of second order for any value of the four-spin interactions. But for relatively high D, the transition temperature depends on the fluctuation of the crystal field, and the system undergoes tricritical behaviour for any strength of the four-spin interactions. We have also found that the model may exhibit reentrance for appropriate values of the system parameters.

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Critical behavior of the Ising model with four-spin coupling

Abstract

J. Phys. C

Phys. Rev. B

J. Phys. A