Journal of Physics A Mathematical and General

Two-dimensional Ising model with multispin interactions

Jean‐Marc Debierre, Loïc Turban

Published Oct 21, 1983• 39 citations

Abstract

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.

DOI: 10.1088/0305-4470/16/15/022

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Journal of Physics A Mathematical and General

Two-dimensional Ising model with multispin interactions

Jean‐Marc Debierre, Loïc Turban

Published Oct 21, 1983• 39 citations

Publisher

Abstract

DOI: 10.1088/0305-4470/16/15/022

Related papers

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Physical review. B, Condensed matterFeb 1, 19908 citations

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.

The spin-1 Ising model on the square lattice with nearest-neighbor ferromagnetic exchange interactions [both bilinear (

J

) and biquadratic (

K

) and crystal-field interaction ($\ensuremath{\Delta}$) is studied via a renormalization-group transformation in position space. The phase diagram in

J

K

, $\ensuremath{\Delta}$ space is found to have one surface of critical phase transitions and two surfaces of first-order phase transitions. These surfaces are variously bounded by an ordinary tricritical line, an isolated critical line, and a line of critical end points. These three lines joint at a special tricritical point corresponding to the transition of the three-state Potts model. The over-all phase diagram is qualitatively similar to that obtained with the mean-field approximation, except in the vicinity of the Potts transition where a four-phase coexistence line in mean-field theory shrinks into a special tricritical point in renormalization-group theory. Symmetry considerations guide the construction of our truncated renormalization-group transformation. The global connectivity and local exponents of the thirteen separate fixed points underlying this quite complicated structure are determined. Local analysis with respect to magnetic field (

H

) and another odd interaction (

L

) is performed. A one-adjusted-parameter version of our transformation yields remarkably quantitative results, predicting the Potts transition temperature, for example, within 0.3% of the exact value.

{T}_{c}

for nonzero uniform magnetic fields H. However, both of the phase boundaries

{T}_{c}

(H) reduce to the exact results available at H=0 and T=0. .AE