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Physical review. B, Condensed matter

Phase transitions in systems with multispin interactions

K. A. Penson, Rémi Jullien, Pierre M. Pfeuty

Published Dec 1, 1982• 78 citations
Publisher

Abstract

An Ising-type model including $n$-spin nearest-neighbor interactions is introduced. The ground-state properties of its Hamiltonian version are studied in one and two dimensions. The model is self-dual in one dimension. The mean-field theory indicates that for $n>{n}_{c}$ the transition may become first order. Finite-size scaling suggests that ${n}_{c}=4$ in one dimension. The critical exponents as functions of $n$ are estimated.

DOI
10.1103/physrevb.26.6334

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An Ising-type model including nnn-spin nearest-neighbor interactions is introduced. The ground-state properties of its Hamiltonian version are studied in one and two dimensions. The model is self-dual in one dimension. The mean-field theory indicates that for n>ncn>{n}_{c}n>nc​ the transition may become first order. Finite-size scaling suggests that nc=4{n}_{c}=4nc​=4 in one dimension. The critical exponents as functions of nnn are estimated.
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A generalized (D+1D+1D+1)-dimensional anisotropic Ising model with a combination of multispin interactions in its DDD dimensions is introduced. A (1 + 1)-dimensional case (D=1D=1D=1) with competing two- ($\ensuremath{\sim}{J}_{2}$) and four-spin ($\ensuremath{\sim}{J}_{4}$) couplings was analyzed through its one-dimensional Hamiltonian version with the use of the finite-size scaling method. Whereas for small J4J2\frac{{J}_{4}}{{J}_{2}}J2​J4​​ the transition is of the usual Ising type, for sufficiently large J4J2\frac{{J}_{4}}{{J}_{2}}J2​J4​​ a first-order phase transition occurs. A comparison with other models with competing interactions was made.
The two-dimensional Ising model in the rectangular lattice is generalised to include m-spin interactions in one direction and two-spin interactions in the other. This model is self-dual and the critical line is the same as in the conventional Ising model with m=2. The m=3 model is solved in a generalised mean-field approximation where the transition is first order. Using a phenomenological renormalisation group approach, approximate values of the thermal and magnetic exponents y t and y h are obtained for m=3 and m=4. The m=3 results together with symmetry arguments suggest that this model belongs to the same class of universality as the Baxter-Wu and four-state ferromagnetic Potts models. When m=4 the transition is probably first order.
Abstract We study the spin-1/2 Ising chain with multispin interactions K involving the product of m successive spins, for general values of m . Using a change of spin variables the zero-field partition function of a finite chain is obtained for free and periodic boundary conditions and we calculate the two-spin correlation function. When placed in an external field H the system is shown to be self-dual. Using another change of spin variables the one-dimensional Ising model with multispin interactions in a field is mapped onto a zero-field rectangular Ising model with first-neighbour interactions K and H . The 2D system, with size m × N / m , has the topology of a cylinder with helical BC. In the thermodynamic limit N / m → ∞ , m → ∞ , a 2D critical singularity develops on the self-duality line, sinh 2 K sinh 2 H = 1 .
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.