Algebraic Reduction of the Ising Model

We consider the Ising model on a cylindrical lattice of L columns, with fixed-spin boundary conditions on the top and bottom rows. The spontaneous magnetization can be written in terms of partition functions on this lattice. We show how we can use the Clifford algebra of Kaufman to write these parti...

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Bibliographic Details
Published inJournal of statistical physics Vol. 132; no. 6; pp. 959 - 982
Main Author Baxter, R. J.
Format Journal Article
LanguageEnglish
Published Boston Springer US 01.09.2008
Springer Nature B.V
Subjects
Algebra
Boundary conditions
Determinants
Ising model
Magnetization
Mathematical and Computational Physics
Partitions (mathematics)
Physical Chemistry
Physics
Physics and Astronomy
Quantum Physics
Statistical Physics and Dynamical Systems
Theoretical
Statistical mechanics
Lattice models
Transfer matrices
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Summary:We consider the Ising model on a cylindrical lattice of L columns, with fixed-spin boundary conditions on the top and bottom rows. The spontaneous magnetization can be written in terms of partition functions on this lattice. We show how we can use the Clifford algebra of Kaufman to write these partition functions in terms of L by L determinants, and then further reduce them to m by m determinants, where m is approximately L /2. In this form the results can be compared with those of the Ising case of the superintegrable chiral Potts model. They point to a way of calculating the spontaneous magnetization of that more general model algebraically.
ISSN:0022-4715
1572-9613
DOI:10.1007/s10955-008-9587-y