Introduction The planar Ising model has become one of the most important statistical mechanical systems for the study of phase transitions and critical phenomena. Although there are many rigorous results, such as correlation inequalities, Peierls argument and the Yang-Lee circle theorem to name but three [reviewed by Griffiths, 1971], which are dimensionindependent in their validity and which lead to results of considerable interest, only the planar model to date benefits from the added insights which stem ultimately from Onsager's tour de force [Onsager, 1944]. It is not the purpose of this article to enter into a general review – for this the reader is referred elsewhere [Gallavotti, 1972] but rather to discuss two more mathematical aspects of the development of Onsager's solution. The first item is the Yang-Baxter system of equations for the planar Ising model in zero field with transfer in the (1,1) direction. This work shows that the Clifford-algebraic structure of the exact solution is a natural consequence of the star-triangle equations. The second item is a Fredholm system which turns out to be of crucial importance in understanding surface and interface problems, as well as the pair correlation function.Introduction The planar Ising model has become one of the most important statistical mechanical systems for the study of phase transitions and critical phenomena. Although there are many rigorous results, such as correlation inequalities, Peierls argument and the Yang-Lee circle theorem to name but three [reviewed by Griffiths, 1971], which are dimensionindependent in their validity and which lead to results of considerable interest, only the planar model to date benefits from the added insights which stem ultimately from Onsager's tour de force [Onsager, 1944]. It is not the purpose of this article to enter into a general review – for this the reader is referred elsewhere [Gallavotti, 1972] but rather to discuss two more mathematical aspects of the development of Onsager's solution. The first item is the Yang-Baxter system of equations for the planar Ising model in zero field with transfer in the (1,1) direction. This work shows that the Clifford-algebraic structure of the exact solution is a natural consequence of the star-triangle equations. The second item is a Fredholm system which turns out to be of crucial importance in understanding surface and interface problems, as well as the pair correlation function.

Cambridge University Press eBooksJan 26, 19890 citations

Exact solutions for Ising-model odd-number correlations on planar lattices

Jeremiah H. Barry, T. Tanaka, Mahfuza Khatun +1 more

Simple procedures are first used to obtain exact solutions of highly-localized odd-number Ising correlations on the kagom\'e, square, and honeycomb lattices. To extend these results, a systematic and unifying method is then developed and demonstrated for finding exact solutions of n-site (with n an odd integer) Ising correlations on various planar lattices. The method combines five transformation or mapping theorems and linear-algebraic correlation identities of the triangular Ising model supplemented by a foreknowledge of its spontaneous magnetization and three select triplet correlations. In particular, considering a select seven-site cluster of the triangular Ising model, the knowledge of all its eleven odd-number correlations defined upon this cluster is shown sufficient for determining exactly all honeycomb, decorated-honeycomb, and kagom\'e Ising odd-number correlations upon their correspondingly select 10-, 19-, and 9-site clusters, respectively. The direct applicability of the catenated mapping theorems and relative ease of the calculational procedures are highlighted by the resulting large numbers of multisite correlation solutions (e.g., approximately 80 and 50 for the honeycomb and kagom\'e Ising models, respectively), the large ${\mathit{n}}_{\mathrm{max}}$ values ( ${\mathit{n}}_{\mathrm{max}}$ =9, 9, and 19, respectively, for the honeycomb, kagom\'e, and decorated-honeycomb Ising models), and convenient prescriptions for extracting critical amplitudes. The results also offer examples of correlation degeneracies and other linear-algebraic correlation identities that do not depend explicitly upon the interaction parameters.Simple procedures are first used to obtain exact solutions of highly-localized odd-number Ising correlations on the kagom\'e, square, and honeycomb lattices. To extend these results, a systematic and unifying method is then developed and demonstrated for finding exact solutions of n-site (with n an odd integer) Ising correlations on various planar lattices. The method combines five transformation or mapping theorems and linear-algebraic correlation identities of the triangular Ising model supplemented by a foreknowledge of its spontaneous magnetization and three select triplet correlations. In particular, considering a select seven-site cluster of the triangular Ising model, the knowledge of all its eleven odd-number correlations defined upon this cluster is shown sufficient for determining exactly all honeycomb, decorated-honeycomb, and kagom\'e Ising odd-number correlations upon their correspondingly select 10-, 19-, and 9-site clusters, respectively. The direct applicability of the catenated mapping theorems and relative ease of the calculational procedures are highlighted by the resulting large numbers of multisite correlation solutions (e.g., approximately 80 and 50 for the honeycomb and kagom\'e Ising models, respectively), the large ${\mathit{n}}_{\mathrm{max}}$ values (${\mathit{n}}_{\mathrm{max}}$=9, 9, and 19, respectively, for the honeycomb, kagom\'e, and decorated-honeycomb Ising models), and convenient prescriptions for extracting critical amplitudes. The results also offer examples of correlation degeneracies and other linear-algebraic correlation identities that do not depend explicitly upon the interaction parameters.

Physical review. B, Condensed matterAug 1, 199145 citations

{\mathit{n}}_{\mathrm{max}}

values (

{\mathit{n}}_{\mathrm{max}}

=9, 9, and 19, respectively, for the honeycomb, kagom\'e, and decorated-honeycomb Ising models), and convenient prescriptions for extracting critical amplitudes. The results also offer examples of correlation degeneracies and other linear-algebraic correlation identities that do not depend explicitly upon the interaction parameters.