Spontaneous magnetization of the superintegrable chiral Potts model: calculation of the determinant D_PQ

Details Spontaneous magnetization of the superintegrable chiral Potts model: calculation of the determinant D_PQ Spontaneous magnetization of the superintegrable chiral Potts model: calculation of the determinant D_PQ

2009-12-23

arxiv.org/abs/0912.4549v2

J. Phys. A Math. Theor. 43, 145002 (16 pages) (2010)

Physics

Condensed Matter

Statistical Mechanics

16 pages, no figures; revised 11 Jan 2010 to correct citations and to include reference to subsequent work

Scientific paper

For the Ising model, the calculation of the spontaneous magnetization leads to the problem of evaluating a determinant. Yang did this by calculating the eigenvalues in the large-lattice limit. Montroll, Potts and Ward expressed it as a Toeplitz determinant and used Szego's theorem: this is almost certainly the route originally travelled by Onsager. For the corresponding problem in the superintegrable chiral Potts model, neither approach appears to work: here we show that the determinant D_PQ can be expressed as that of a product of two Cauchy-like matrices. One can then use the elementary exact formula for the Cauchy determinant. One of course regains the known result, originally conjectured in 1989.

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