Algebraic reduction of the Ising model

Abstract

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.

25 pages, one figure, last reference completed. Various typos fixed. Changes on 12 July 2008: Fig 1, 0 to +1; before (2.1), if to is; after (4.6), from to form; before (4.46), first three to middle two; before (4.46), last to others; Conclusions, 2nd para, insert how ; renewcommand ıto be \rm i

References in corpus (3)

• Derivation of the order parameter of the chiral Potts model
• The order parameter of the chiral Potts model
• A conjecture for the superintegrable chiral Potts model

Cited by in corpus (6)

• On the form factors of local operators in the Bazhanov-Stroganov and chiral Potts models
• The tau_2-model and the chiral Potts model revisited: completeness of Bethe equations from Sklyanin's SOV method
• Spontaneous magnetization of the superintegrable chiral Potts model: calculation of the determinant D_PQ
• A conjecture for the superintegrable chiral Potts model
• Spontaneous Magnetization of the Integrable Chiral Potts Model
• Some comments on developments in exact solutions in statistical mechanics since 1944