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Acta Physica Sinica

AN APPROXIMATE SOLUTION FOR SQUARE-LATTICE SPIN-1 MODEL

YE QING, 上海交通大学应用物理系凝聚态物理研究所

Published Jan 1, 1989• 0 citations
Publisher

Abstract

In this paper, we apply the decoration and decimation transformation to the square-lattice spin-1 model and show that it is equivalent toa checkerboard Ising model with cross and four-spin interactions. Then, the approximate critical condition is obtained for the model and it reduces to Onsager's exact solution in the Ising-model limit.

DOI
10.7498/aps.38.1187

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Physical review. B, Condensed matter•Oct 1, 1979•13 citations
In this paper, we apply the decoration and decimation transformation to the square-lattice spin-1 model and show that it is equivalent toa checkerboard Ising model with cross and four-spin interactions. Then, the approximate critical condition is obtained for the model and it reduces to Onsager's exact solution in the Ising-model limit.
Using the decimation technique,a decorated square Ising lattice with two kinds of interactions K1and K2 was transformed into a regular square Ising lattice with nearest-neighbor(NN),next-nearest-neighbor(NNN),and four-spin interactions.The critical point of the approximate solution in this system was at K′C=0.4406868.The result indicated that the phase transition occurred only when the NN interaction K1 and the NNN interaction K2 satisfied a certain relation.If we only calculated the NN interaction K1,the critical point of the exact solution was at K1C=0.7635.It was also found that if we decorated two spins between the two square lattice points,the answer could be solved exactly on condition that the NN interaction existed on the doubly decorated square Ising lattice.
A decorated lattice is suggested and the Ising model on it with three kinds of interactions K1, K2, and K3 is studied. Using an equivalent transformation, the square decorated Ising lattice is transformed into a regular square Ising lattice with nearest-neighbor, next-nearest-neighbor, and four-spin interactions, and the critical fixed point is found at K1 = 0.5769, K2 = −0.0671, and K3 = 0.3428, which determines the critical temperature of the system. It is also found that this system and the regular square Ising lattice, and the eight-vertex model belong to the same universality class.
An exact duality-decimation transformation is derived for a large class of two-dimensional spin systems on a square lattice. It maps a model with interactions in every second square (checkerboard interactions) onto a model with interactions in every square and vice versa. The transformation is applicable to Ising, Potts, solid-on-solid (SOS), and $x\ensuremath{-}y$ models. It may be used to reformulate the site percolation problem in terms of the $s\ensuremath{\rightarrow}1$ limit of a Potts model with four-spin interactions in every elementary square. The transformation maps the van Beijeren SOS model, which is equivalent to the FFF model, onto an $x\ensuremath{-}y$ model with checkerboard interactions. Thus one has an exactly soluble $x\ensuremath{-}y$ model, which, however, bears little resemblance to the ordinary $x\ensuremath{-}y$ model. The transformation may be combined with a bond-shifting approximation to obtain a real-space renormalization scheme which in some respects represents an improvement over the Migdal-Kadanoff formula. No nearest-neighbor bonds are shifted. Better values of the exponent $\ensuremath{\nu}$ are obtained for the Ising and Potts models. Apart from exponentially vanishing corrections, the renormalization transformation for SOS and $x\ensuremath{-}y$ models has a surface of fixed points corresponding to generalizations of the usual discrete Gaussian and Villain models.