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CHINESE JOURNAL OF PHYSKS VOL. 27. NO. 3 JUNE 1989

Exact Solutions of the lsing Model on a Triangular Checkerboard

Lattice with Crossing interactions

K.Y.Lin(j$%@)

Physics Department, National Tsing Hua University, Hsinchu

(Received 6, Dec. 1988)

We consider the Ising model on a triangular checkerboard latticewhere the internal structure of the unit cell is arbitrary and crossing

interactions are allowed. The spontaneous magnetization of the

internal spin within a unit cell is derived.

It is well known that the free energy of the Ising model on any two-dimensional lattice

with noncrossing interactions can be derived exactly by the method of Pfaffian.’ Similarly

the spontaneous magnetization on various planar lattices with noncrossing interactions have

been derived.’ ,2 However exact results for two-dimensional lattices with crossing inter-

actions are still very rare.

The purpose of this paper is to derive exact solutions of the Ising model ona triangular

checkerboard lattice. The interactions inside a unit cell is assumed to be invariant under the

operation of the total spin reversion. Crossing interactions are allowed. The spontaneous

magnetization of the internal spin within a unit cell is derived. Our result is given in terms

of Boltzmann weights of a unit cell without specifying its cell structures.

Consider the triangular checkerboard Ising lattice as shown in Fig. 1. Each shaded

FIG. 1 The triangular checkerboard lattice.

21 2



K. Y. LIN 217

FIG. 4 A unit cell consisting of one A network and n Y networks.

pointing) triangular network is equivalent to a triangle with three pairwise interactions J, ,

J2, J3 (J ’ , , S, , J ’ , ) as shown in Fig. 5. The spontaneous magnetization of the internal spin

X within the up-pointing triangular network is given by (14). The one-spin and three-spin

FIG. 5 A Kagome lattice.

correlations of the nodal spins for the Kagome lattice Ising model have been derived:’ ._ ”

recently by Lin and Chen.’ Their result can be summarized as follows: The triangle

network with three interactions Ji(Ji), i = 1,2,3, can be transformed into a star network with

three interactions L,(Li), i = 1,2,3, by the well known star-triangle transformation2 It is

shown by Lin and Chen that

<ai> = tanh(li + Li) M, (18)

where M, is the spontaneous magnetization of the honeycomb lattice with three interac-

tions K, , Kz, K, such that

tanhy. = tanhLitanhLi . (19)

The three-spin correlation is given by

L-. .._ .-



318 EXACT SOLUTIONS OF THE lSING MODEL ON A TRIANGULAR CHECKERBOARD

LATTICE WITH CROSSING INTERACTIONS

<a, a:a3>lMo = [ 1 - a, tanh(L, + L; ) - a2 tanh(L2 + L; )

-ua3tanh(L, +L;)]/b . (20)

The author thanks Prof. R. J. Baxter and Dr. T. C. Choy for sending their preprint

prior to publication. This research is supported by the National Science Council of the

Republic of China.

REFERENCES

1. B. M. McCoy and T. T. Wu, The Two-Dimensional Ising Model (Harvard University

Press, Cambridge, 1973).

2. I. Syozi, Phase Transition and Critical Phinomena, Vol. 1, edgs. C. Domb and M. S.

Green (Academic, London, 1972) p. 269.

. 3. M. E. Fisher, Phys. Rev. 113, 969 (1959).

4. R. J. Baxter, J. Phys. AS, 1979 (1975).

5. I. G. Enting, J. Phys. AlO, 1737 (1977).

6. R. J. Baxter and T. C. Choy, Proc. Roy. Sot. A423, 279 (1989).

7. T. C. Choy and R. J. Baxter, Phys. Lett. A125, 365 (1987).

8. K. Y. Lin and F. Y. Wu, J. Phys. A22, 1121 (1989).

9. K. Y. Lin and B. H. Chen, Int. J. Mod. Phys. B (submitted).

k.y. lin- exact solutions of the lsing model on a triangular checkerboard lattice with crossing...

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