Progress of Theoretical Physics, Vol. VI, No.3, May-June, 1951. Statistics of Kagome Lattice

The transition temperature of the kagome lattice with Z=4 is obtained and compared with that of the square lattice. After the work of Onsager,I) who solved exactly the problem of Ising model for the case of plane square lattice, the same problems for the honeycomb and triangular lattice were treated by several authors.2) Other than these three types of lattices, there is left a lattice, called in Japanese kagome (woven bomboo pattern), which consists exclusively of equivalent lattice points and equivalent bonds. Since the number of nearest neighbors of a lattice point is as many as in the square lattice, namely four, it is interesting to verify the natural conjecture that the curie point, in general, is determined solely by the relation ch2H = sec 7< / Z established by Onsager for the three types of lattices. Fig ' 1. Kagome Lattice Let us stat t from a variant of the honeycomb lattice, which has an extra spin on the middle point of every side as well as on every vertex (say decorated honeycomb lattice). Let its interaction parameter be L. By summing at first over the spin variables with respect to the vertices in the partition function of this lattice, we arrive at the partition function of the kagome lattice (Star-triangle transformation), with an interaction parameter K; in fin