Abstract
The two-dimensional Ising model for a system of interacting spins (or for the ordering of an AB alloy) on a square lattice is one of the very few nontrivial many-body problems that is exactly soluble and shows a phase transition. Although the exact solution in the absence of an external magnetic field was first given almost twenty years ago in a famous paper by Onsager1 using the theory of Lie algebras, the flow of papers on both approximate and exact methods has remained strong to this day.2 One reason for this has been the interest in testing approximate methods on an exactly soluble problem. A second reason, no doubt, has been the considerable formidability of the Onsager method. The simplification achieved by Bruria Kaufman3 using the theory of spinor representations has diminished, but not removed, the reputation of the Onsager approach for incomprehensibility, while the subsequent application of this method by Yang4 to the calculation of the spontaneous magnetization has, if anything, helped to restore this reputation.
Present address: Belfer Graduate School of Science, Yeshiva University, New York, New York.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Reference
L. Onsager, Phys. Rev. 65, 117 (1944).
For reviews, see G. F. Newell and E. W. Montroll, Rev. Mod. Phys. 25, 353 (1953); and C. Domb, Phil. Mag. Suppl. 9, 151 (1960).
B. Kaufman, Phys. Rev. 76, 1232 (1949); B. Kaufman and L. Onsager, Phys. Rev. 76, 1244 (1949). See also Y. Nambu, Progr. Theoret. Phys. (Kyoto) 5, 1 (1950); K. Husimi and I., Syózi, Progr. Theoret. Phys. (Kyoto) 5, 177 (1950); and L Syózi Progr. Theoret. Phys. (Kyoto) 5, 341 (1950).
C. N. Yang, Phys. Rev. 85, 808 (1952).
B. L. van der Waerden, Z. Physik 118, 473 (1941).
M. Kac and J. C. Ward, Phys. Rev. 88, 1332 (1952). See also S. Sherman, J. Math, Phys. 1, 202 (1960); and P. N. Burgoyne, J. Math. Phys. 4, 1320 (1963) who have supplied proofs necessary to make the approach of Kac and Ward rigorous.
R. B. Potts and J. C. Ward, Progr. Theoret Phys. (Kyoto) 13, 38 (1955).
C. A. Hurst and H. S. Green, J. Chem. Phys. 33, 1059 (1960). See also A. M. Dykhne and Yu. B. Rumer, Usp. Fiz. Nauk 75, 101 (1961) [English transl.: Soviet Phys.—Usp. 4, 698 (1962)].
P. W. Kasteleyn, J. Math. Phys. 4, 287 (1963).
E. W. Montroll, R. B. Potts, and J. C. Ward, J. Math, Phys. 4, 308 (1963).
N. N. Bogolubov, Nuovo Cimento 7, 794 (1958).
J. G. Valatin, Nuovo Cimento 7, 843 (1958).
See the review of Newell and Montroll, Ref. 2. This and related methods were originally discovered by E. W. Montroll, J. Chem. Phys. 9, 706 (1941);
H. A. Kramers and G. H. Wannier, Phys. Rev. 60, 252 (1941);
E. N. Lassettre and J. P. Howe, J. Chem. Phys. 9, 747 (1941); and R. Kubo, Busseiron Kenkyu 1, 1943.
H. Schmidt, Phys. Rev, 105, 425 (1957).
J. Bardeen, L. Cooper, and J. Schrieffer, Phys. Rev. 108, 1175 (1957).
P. W. Anderson, Phys. Rev. 112, 1900 (1958).
See remark of Domb, Ref. 2, p. 194.
J. Ashkin and W. E. Lamb, Phys. Rev. 64, 159 (1943).
K.. Huang, Statistical Mechanics (John Wiley & Sons, Inc., New York, London, 1963), pp. 369 ff.
This situation is identical to;rhat we found for the “XY model” of quantum mechanical spins in one dimension, where it is discussed in more detail. See E. Lieb, D. Mattis, and T. Schultz, Ann. Phys. 16, 407 (1961), especially pp. 417 ff.
See V. Grenander and G. Szegö, Toeplits Forms and their Applications (University of California Press, Berkeley, Cali¬fornia, 1958 ). For a proof that is extendable to non-Hermitian kernels, as is the case here, see M. Kac, Probability and Related Topics in Physical Sciences (Interscience Publishers, Inc., New York, 1959 ).
Dr. Montroll and Dr. Potts have kindly informed us that their use of the Szegö-Kac theorem in this case is not correct as it appears in MPW, but that it can be corrected.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Schultz, T.D., Mattis, D.C., Lieb, E.H. (2004). Two-Dimensional Ising Model as a Soluble Problem of Many Fermions. In: Nachtergaele, B., Solovej, J.P., Yngvason, J. (eds) Condensed Matter Physics and Exactly Soluble Models. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-06390-3_33
Download citation
DOI: https://doi.org/10.1007/978-3-662-06390-3_33
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-06093-9
Online ISBN: 978-3-662-06390-3
eBook Packages: Springer Book Archive