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Journal of Physics C Solid State Physics

Magnetic-field-induced phase transition in a model with three-spin interactions

K Jungling, Gustav M. Obermair

Published Feb 7, 1975• 4 citations
Publisher

Abstract

The authors transform a two-dimensional model of Ising spins with pure three-spin interactions and a special magnetic field into a spin model which is equivalent to the Baxter model. This shows that the original model exhibits a phase transition which is induced by the field h and whose critical exponents depend continuously on the strength of h.

DOI
10.1088/0022-3719/8/3/001

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The authors transform a two-dimensional model of Ising spins with pure three-spin interactions and a special magnetic field into a spin model which is equivalent to the Baxter model. This shows that the original model exhibits a phase transition which is induced by the field h and whose critical exponents depend continuously on the strength of h.
The equilibrium and phase properties of the Ising model with three-spin interaction and an external field are studied within the framework of mean-field approximation. The thermodynamic properties of the model reveals two coexistence curves, signifying two distinct second-order phase transitions, dependent on the domain of the interaction parameter. The critical exponents of the magnetic order parameter are calculated in all directions of the phase space and show their agreement with the mean-field universality class.
The three-spin interaction Hamiltonian ${\mathcal{H}}^{(3)}\ensuremath{\equiv}\ensuremath{-}{J}_{3}\ensuremath{\Sigma}{〈\mathrm{ijk}〉}^{}{s}_{i}{\ensuremath{\sigma}}_{j}{\ensuremath{\sigma}}_{k}\ensuremath{-}H\ensuremath{\Sigma}{i}^{}{s}_{i}\ensuremath{-}{H}^{\ensuremath{'}}\ensuremath{\Sigma}{j}^{}{\ensuremath{\sigma}}_{j}$ (the sss and $\ensuremath{\sigma}$ spins belong to different sublattices) is solved for some two-dimensional lattices by a generalized star-triangle transformation. The ${H}^{\ensuremath{'}}=0$ internal energy, specific heat, and magnetization are explicitly calculated, and the singularity structures of other functions (e.g.xxx are studied. Although the actual critical-point exponents remain the same as for systems with two-spin interaction, the three-spin interaction gives rise to different amplitude and background terms. In particular, the ${H}^{\ensuremath{'}}=0$ magnetization is nonzero for all finite TTT, because of the lack of spin-reversal symmetry; the diameter is seen explicitly to have an energy singlularity, with critical exponent $1\ensuremath{-}\ensuremath{\alpha}$.
Extensive Monte Carlo simulations have been performed to study the effect of an applied magnetic field in a two-dimensional system of XY\mathrm{XY}XY spins interacting with each other via ferromagnetic and antiferromagnetic bonds. The magnetic field H reduces the system symmetry O(2) in spin space to Z2{Z}_{2}Z2​. We show that there exists a critical value of H below which an Ising-like transition occurs. However, there remain continuous degrees of freedom so that critical exponents deviate from those of the pure Ising model.
The one-dimensional spin-\textonehalf{} Ising model is solved exactly with very long-range, equivalent-neighbor ferromagnetic interactions of strength JLR{J}_{\mathrm{LR}}JLR​ superimposed on nearest-neighbor anti-ferromagnetic interactions of strength $\ensuremath{-}{J}_{\mathrm{sr}}$. For $\ensuremath{-}\frac{{J}_{\mathrm{lr}}}{{J}_{\mathrm{sr}}} >R=3.1532$... there is a critical point of classical type in zero field. For $2\ensuremath{-}\frac{{J}_{\mathrm{lr}}}{{J}_{\mathrm{sr}}}=R$ the critical exponents take the values $\ensuremath{\beta}=\frac{1}{4}$, $\ensuremath{\gamma}=1={\ensuremath{\gamma}}^{\ensuremath{'}}$, $\ensuremath{\delta}=5$, and $0=\ensuremath{\alpha}\ensuremath{\ne}{\ensuremath{\alpha}}^{\ensuremath{'}}=\frac{1}{2}$. For $2<\ensuremath{-}\frac{{J}_{\mathrm{lr}}}{{J}_{\mathrm{sr}}}<R$ the transition in zero field is of first order, and the critical points which occur in finite field are of classical type. For $0<\ensuremath{-}\frac{{J}_{\mathrm{lr}}}{{J}_{\mathrm{sr}}}<2$ there is no phase transition in zero field, although there are classical critical points in a field.
In the framework of mean field approximation, we consider a spin system consisting of two interacting sub-ensembles. The intra-ensemble interactions are ferromagnetic, while the inter-ensemble interactions are antiferromagnetic. We define the effective number of the nearest neighbors and show that if the two sub-ensembles have the same effective number of the nearest neighbors, the classical form of critical exponents (α=0, β=1/2, γ=γ'=1, δ=3) gives way to the non-classical form (α=0, β=3/2, γ=γ'=0, δ=1), and the scaling function changes simultaneously. We demonstrate that this system allows for two second-order phase transitions and two first-order phase transitions. We observe that an external magnetic field does not destroy the phase transitions but only shifts their critical points, allowing for control of the system's parameters. We discuss the regime when the magnetization as a function of the magnetic field develops a low-magnetization plateau and show that the height of this plateau abruptly rises to the value of one when the magnetic field reaches a critical value. Our analytical results are supported by a Monte Carlo simulation of a three-dimensional layered model.