No data available.
Please log in to see this content.
You have no subscription access to this content.
No metrics data to plot.
The attempt to load metrics for this article has failed.
The attempt to plot a graph for these metrics has failed.
Correlations and Spontaneous Magnetization of the Two‐Dimensional Ising Model
1.L. Onsager, Nuovo Cimento, Suppl. 6, 261 (1949).
2.Onsager’s celebrated derivation of the partition function of the two‐dimensional model is in Phys. Rev. 65, 117 (1944).
3.C. N. Yang, Phys. Rev. 85, 808 (1952).
3.See also C. H. Chang, Phys. Rev. 88, 1422 (1952), where the result for the rectangular lattice (different vertical and horizontal interactions) is obtained.
4.B. Kaufman and L. Onsager, Phys. Rev. 76, 1244 (1949).
5.R. B. Potts and J. C. Ward, Progr. Theoret. Phys. (Kyoto) 13, 38 (1955).
6.P. W. Kasteleyn, J. Math. Phys. 4, 287 (1963).
7.M. Kac and J. C. Ward, Phys. Rev. 88, 1332 (1952).
8.C. A. Hurst and H. S. Green, J. Chem. Phys. 33, 1059 (1960).
9.See also A. M. Dykhne and Yu. B. Rumer, Soviet Phys‐Usp. (English transl.) 75, 698 (1962).
10.E. W. Montroll, “Lattice Statistics,” to appear as a chapter in a book entitled Applied Combinatorial Mathematics, edited by E. F. Beckenbach.
11.G. Szego, “On Certain Hermitian forms associated with the Fourier series of a positive function,” Communications du seminaire mathematique de l’université de Lund, tome supplementaire (1952) dedié a Marcel Riesz p. 228–238.
11.See also V. Grenander and G. Szego, Toeplitz Forms and their Applications (University of California Press, Berkeley, California, 1958),
11.as well as M. Kac, Duke Math. J. 21 (1954).
12.In general, the notation of this paper will follow that of the review article by Newell and Montroll. The variables and are chosen instead of K and to avoid an inconsistency in the use of this notation in the review article and in Kaufman’s paper. To compare our results with Kaufman and Onsager’s it is necessary to take and and to compare with the results of Potts and Ward, take and
13.G. F. Newell and E. W. Montroll, Rev. Mod. Phys. 25, 353 (1953).
14.B. Kaufman, Phys. Rev. 76, 1232 (1949).
15.The importance of these identities in the simplification of the Potts and Ward analysis of correlations has been observed independently by H. S. Green. While this paper was being prepared, we received a copy of a manuscript by H. S. Green and C. A. Hurst on the same subject. Although their treatment of correlation is similar to ours they have not taken advantage of the theorems on Toeplitz forms which are so helpful in spontaneous magnetization calculations. Their paper is to be published in a Max Born Festschrift issue of Z. Physik.
16.There is more than one rule for evaluating Pfaffians. Perhaps the mathematical physicist should prefer that suggested by Thomson and Tait in their Treatise on Natural Philosophy (Cambridge University Press, Cambridge, England (1879).
16.In the 1962 Dover republication under the title Principles of Mechanics and Dynamics, the rule appears on p. 394 of Part I.
16.See also E. Caianello, Nuovo Cimento, Suppl. 14, 177 (1959).
17.Actually Eq. (68) has not appeared in the literature. The work of Szego 11 is restricted to the case of Hermitian Toeplitz forms so that the of (68) appears as in his paper. However, the methods discussed in the Kac article can be applied immediately in the general case to derive (68).
18.A. Erdelyi, W. Magnus, F. Oberhettinger, and F. Tricomi, Tables of Integral Transforms (McGraw Hill Book Company, Inc., New York, 1954), Vol. 1, p. 183.
19.E. W. Montroll, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability (University of California Press, 1956), Vol. III, p. 209.
20.A. A. Maradudin, E. W. Montroll, G. H. Weiss, R. Herman, and H. L. Milnes, Mem. Acad. Prog. de Belgique XIV, No. 1709 (1960).
21.Equation (43) of their paper contains a misprinted sign. To change from their notation to ours note that
Article metrics loading...
Full text loading...