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Physical review. B, Solid state

Ising Model with Four-Spin Interactions

Fa Yueh Wu

Published Feb 1, 1972• 11 citations
Publisher

Abstract

It is shown that Baxter's recent results on a lattice-statistical model lead to the solution of an Ising model with two- and four-spin interactions. Critical properties of this Ising model in various regions of the parameter space are given. It is argued that four-spin or crossing interactions in a two-dimensional Ising model would in general lead to a critical exponent ${\ensuremath{\alpha}}^{\ensuremath{'}}\ensuremath{\ne}0$.

DOI
10.1103/physrevb.5.1186

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Leo P. Kadanoff, Franz Wegner

The eight-vertex model solved by Baxter is shown to be equivalent to two Ising models with nearest-neighbor coupling interacting with one another via a four-spin coupling term. The critical properties of the model in the weak-coupling limit are in agreement with the scaling hypothesis. In this limit where $\ensuremath{\alpha}\ensuremath{\rightarrow}0$, the critical indices obey $\frac{\ensuremath{\gamma}}{{\ensuremath{\gamma}}_{0}}=\frac{\ensuremath{\beta}}{{\ensuremath{\beta}}_{0}}=\frac{\ensuremath{\nu}}{{\ensuremath{\nu}}_{0}}=1\ensuremath{-}\frac{1}{2}\ensuremath{\alpha}$, $\frac{\ensuremath{\delta}}{{\ensuremath{\delta}}_{0}}=\frac{\ensuremath{\eta}}{{\ensuremath{\eta}}_{0}}=1$, with the subscripts zero denoting the index values for the ordinary two-dimensional Ising model.

Physical review. B, Solid state•Dec 1, 1971•373 citations
It is shown that Baxter's recent results on a lattice-statistical model lead to the solution of an Ising model with two- and four-spin interactions. Critical properties of this Ising model in various regions of the parameter space are given. It is argued that four-spin or crossing interactions in a two-dimensional Ising model would in general lead to a critical exponent \ensuremathα\ensuremath′\ensuremath≠0{\ensuremath{\alpha}}^{\ensuremath{'}}\ensuremath{\ne}0\ensuremathα\ensuremath′\ensuremath=0.
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.
The eight-vertex model solved by Baxter is shown to be equivalent to two Ising models with nearest-neighbor coupling interacting with one another via a four-spin coupling term. The critical properties of the model in the weak-coupling limit are in agreement with the scaling hypothesis. In this limit where $\ensuremath{\alpha}\ensuremath{\rightarrow}0$, the critical indices obey $\frac{\ensuremath{\gamma}}{{\ensuremath{\gamma}}_{0}}=\frac{\ensuremath{\beta}}{{\ensuremath{\beta}}_{0}}=\frac{\ensuremath{\nu}}{{\ensuremath{\nu}}_{0}}=1\ensuremath{-}\frac{1}{2}\ensuremath{\alpha}$, $\frac{\ensuremath{\delta}}{{\ensuremath{\delta}}_{0}}=\frac{\ensuremath{\eta}}{{\ensuremath{\eta}}_{0}}=1$, with the subscripts zero denoting the index values for the ordinary two-dimensional Ising model.