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Journal of Physics A Mathematical and General

Critical properties of an S=1 multispin coupling Ising model

Ferenc Iglói, D. Kapor, M. J. Škrinjaŕ

Published Mar 11, 1987• 2 citations
Publisher

Abstract

Strong and weak coupling series expansions are used to investigate the critical properties of the S=1 multispin coupling Ising chain in the presence of a transverse field. The order of the transition goes from second order to first order when more than three spins are coupled. According to the numerical results the two-spin and three-spin coupling models belong to the Ising and the four-state Potts universality classes, respectively. Thus the critical properties of the model do not depend on the value of the spin, even when three spins are coupled.

DOI
10.1088/0305-4470/20/4/012

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Strong and weak coupling series expansions are used to investigate the critical properties of the S=1 multispin coupling Ising chain in the presence of a transverse field. The order of the transition goes from second order to first order when more than three spins are coupled. According to the numerical results the two-spin and three-spin coupling models belong to the Ising and the four-state Potts universality classes, respectively. Thus the critical properties of the model do not depend on the value of the spin, even when three spins are coupled.
We apply a real-space block renormalization group approach to study the critical properties of the random transverse-field Ising spin chain with multispin interactions. First, we recover the known properties of the traditional model with two-spin interactions by applying the renormalization approach for the arbitrary size of the block. For the model with three-spin couplings, we calculate the critical point and demonstrate that the phase transition is controlled by an infinite disorder fixed point. We have determined the typical correlation-length critical exponent, which seems to be different from that of the random transverse Ising chain with nearest-neighbor couplings. Thus, this model represents a new infinite disorder universality class.
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.
Both weak-coupling and strong-coupling series expansions are calculated for the Ising model in a transverse field at zero temperature in three dimensions. Series are obtained for the ground-state energy, the magnetization, the susceptibility, and the energy gaps, on the simple cubic, the body-centred cubic and the face-centred cubic lattices. The analysis of the critical behaviour is consistent with the behaviour predicted by renormalization group theory for the four-dimensional simple Ising model. There is a remarkable degree of universality between all three lattices.
We explore the phase diagram of Ising spins on one-dimensional chains that criss-cross in two perpendicular directions and that are connected by interchain couplings. This system is of interest as a simpler, classical analog of a quantum Hamiltonian that has been proposed as a model of magnetic behavior in Nb_{12}O_{29} and also, conceptually, as a geometry that is intermediate between one and two dimensions. Using mean-field theory as well as Metropolis Monte Carlo and Wang-Landau simulations, we locate quantitatively the boundaries of four ordered phases. Each becomes an effective Ising model with unique effective couplings at large interchain coupling. Away from this limit, we demonstrate nontrivial critical behavior, including tricritical points that separate first- and second-order phase transitions. Finally, we present evidence that this model belongs to the two-dimensional Ising universality class.