Proof of the determinantal form of the spontaneous magnetization of the superintegrable chiral Potts model

Abstract

The superintegrable chiral Potts model has many resemblances to the Ising model, so it is natural to look for algebraic properties similar to those found for the Ising model by Onsager, Kaufman and Yang. The spontaneous magnetization $\mathcal{M}_r$ can be written in terms of a sum over the elements of a matrix $S_r$. The author conjectured the form of the elements, and this conjecture has been verified by Iorgov et al. The author also conjectured in 2008 that this sum could be expressed as a determinant, and has recently evaluated the determinant to obtain the known result for $\mathcal{M}_r$. Here we prove that the sum and the determinant are indeed identical expressions. Since the order parameters of the superintegrable chiral Potts model are also those of the more general solvable chiral Potts model, this completes the algebraic calculation of $\mathcal{M}_r$ for the general model. doi:10.1017/S1446181110000787