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Physical review. B, Condensed matter

Exact solutions for even-number correlations of the square Ising model

Mahfuza Khatun, Jeremiah H. Barry, T. Tanaka

Published Sep 1, 1990• 39 citations
Publisher

Abstract

To obtain an unusually large set of exact solutions for even-number localized correlations of the standard square Ising model, a method is developed that combines traditional Pfaffian techniques with linear-algebraic systems of correlation identities having interaction-dependent coefficients. Two eight-site clusters (subclusters of the ``Greek cross'' cluster) are studied, and altogether sixty different even-number correlations are determined exactly at all temperatures. Besides demonstrating the existence of degeneracies within the set of exact solutions, the solution curve for an eight-site correlation is evidently the first of such large order to be displayed for an Ising model on any regular planar lattice. Significant time as well as labor efficiency of the method is clearly demonstrated since relatively few correlations need to be actually calculated by Pfaffian techniques in order to obtain large additional numbers of multisite correlation solutions using much simpler linear-algebraic procedures.

DOI
10.1103/physrevb.42.4398

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To obtain an unusually large set of exact solutions for even-number localized correlations of the standard square Ising model, a method is developed that combines traditional Pfaffian techniques with linear-algebraic systems of correlation identities having interaction-dependent coefficients. Two eight-site clusters (subclusters of the ``Greek cross'' cluster) are studied, and altogether sixty different even-number correlations are determined exactly at all temperatures. Besides demonstrating the existence of degeneracies within the set of exact solutions, the solution curve for an eight-site correlation is evidently the first of such large order to be displayed for an Ising model on any regular planar lattice. Significant time as well as labor efficiency of the method is clearly demonstrated since relatively few correlations need to be actually calculated by Pfaffian techniques in order to obtain large additional numbers of multisite correlation solutions using much simpler linear-algebraic procedures.
A systematic and unifying method is developed and demonstrated for obtaining exact solutions of n-site (n even integer) Ising correlations on various planar lattices. The scheme, which is exceedingly more simple than using solely traditional Pfaffian techniques, embodies five mapping theorems in alliance with algebraic correlation identities. In the theoretical framework, the triangular Ising model plays an overarching role. In particular, considering a select 7-site cluster of the triangular Ising model, the knowledge of all its 11 even-number correlations defined upon this cluster (where only four of the correlations need to be actually calculated by Pfaffian procedures) is shown to be sufficient for determining exactly all honeycomb, decorated-honeycomb, and kagom\'e Ising model even-number correlations upon their correspondingly select 10-site, 19-site, and 9-site clusters, respectively. The relative ease and direct applicability of the present approach are highlighted not only by the resulting large numbers of n-site (n even-integer) correlation solutions (e.g., approximately 85 and 50 for the honeycomb and kagom\'e Ising models, respectively) and their large nmax{n}_{\mathrm{max}}nmax​ values (nmax=8{n}_{\mathrm{max}=8}nmax=8​,10,18 for the kagom\'e, honeycomb, and decorated-honeycomb Ising models, respectively) but also by the realization that the exact solutions for Ising multisite correlations upon the kagom\'e lattice (one of the four regular lattices in two dimensions) are apparently the first to explicitly appear in the literature beyond its nearest-neighbor pair correlation (energy).
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 nmax{\mathit{n}}_{\mathrm{max}}nmax​ values (nmax{\mathit{n}}_{\mathrm{max}}nmax​=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.
The 3-12 (or extended kagom\'e) lattice is a three-coordinated irregular planar lattice having physical applications. Viewing its sites as the decoration sites of a doubly decorated honeycomb lattice, one proves via local star-triangle and double decoration-decimation transformations that 3-12 Ising correlations can be conveniently represented as linear combinations of honeycomb Ising correlations. Existent knowledge of all honeycomb Ising correlations upon a select (spatially compact) 10-site cluster is thus sufficient to determine all 3-12 Ising correlations upon an associated 18-site cluster. The total number of 3-12 Ising correlations defined upon this 18-site cluster is exceedingly large, but their actual count is less significant than the realization that each can now be found in a systematic and efficient fashion. Examples of resulting exact solutions for both even- and odd-number multisite correlations of the 3-12 Ising ferromagnet are presented at all temperatures. A simple scaling relationship is established between the asymptotic forms of the pair correlation in the 3-12 and honeycomb Ising models. Besides providing relatively direct derivations (no explicit magnetic fields or field derivatives) for the spontaneous magnetization and internal energy of the 3-12 Ising model, the mapping methods may be repeated recursively to secure Ising multisite correlations upon various other irregular planar lattices.
This thesis project involved the study of exact solutions for magnetic spin correlations on a regular- and a decorated-square Ising lattice structure. Existent knowledge of correlations on a regular-square Ising structure is sufficient to determine exact correlation values on a select, spatially compact, decorated structure. These values can be represented as linear combinations of regular-square Ising spin correlation through the use of linear-algebra and the decoration-iteration transformation. Multispin localized correlations are studied over a broad range of temperatures and are found to decrease with increasing temperature. At very low temperatures, all correlation are found to saturate to unity regardless of spin separation and quantity of spins involved.
Any planar set of intersecting straight lines forms a four-coordinated graph, or 'lattice', provided no three lines intersect at a point. For any such lattice an eight-vertex model can be constructed. Provided the interactions satisfy certain constraints (which are in general temperature-dependent), the model can be solved exactly in the thermodynamic limit, its local properties at a particular site being those of a related square lattice. A particular case is a solvable model on the Kagome lattice. It is shown that this model includes as special cases many of the models in statistical mechanics that have been solved exactly, notably the square, triangular and honeycomb Ising models, and the square eight-vertex model. Some remarkable equivalences between correlations on different lattices are also established.
Abstract Any planar set of intersecting straight lines forms a four-coordinated graph, or ‘lattice’, provided no three lines intersect at a point. For any such lattice an eight-vertex model can be constructed. Provided the interactions satisfy certain constraints (which are in general temperature-dependent), the model can be solved exactly in the thermodynamic limit, its local properties at a particular site being those of a related square lattice. A particular case is a solvable model on the Kagomé lattice. It is shown that this model includes as special cases many of the models in statistical mechanics that have been solved exactly, notably the square, triangular and honeycomb Ising models, and the square eight-vertex model. Some remarkable equivalences between correlations on different lattices are also established.