The challenge of the chiral Potts model

The chiral Potts model continues to pose particular challenges in statistical mechanics: it is ``exactly solvable'' in the sense that it satisfies the Yang-Baxter relation, but actually obtaining the solution is not easy. Its free energy was calculated in 1988 and the order parameter was conjectured in full generality a year later. However, a derivation of that conjecture had to wait until 2005. Here we discuss that derivation.Comment: 22 pages, 3 figures, 29 reference

Derivation of the order parameter of the chiral Potts model

We derive the order parameter of the chiral Potts model, using the method of Jimbo et al. The result agrees with previous conjectures.Comment: Version 2 submitted 21 Feb 2005. It has 7 pages, 2 figures. The introduction has been expanded and a significant typographical error in eqn 23 has been correcte

The order parameter of the chiral Potts model

An outstanding problem in statistical mechanics is the order parameter of the chiral Potts model. An elegant conjecture for this was made in 1983. It has since been successfully tested against series expansions, but as far as the author is aware there is as yet no proof of the conjecture. Here we show that if one makes a certain analyticity assumption similar to that used to derive the free energy, then one can indeed verify the conjecture. The method is based on the ``broken rapidity line'' approach pioneered by Jimbo, Miwa and Nakayashiki.Comment: 29 pages, 7 figures. Citations made more explicit and some typos correcte

Corner transfer matrices in statistical mechanics

Corner transfer matrices are a useful tool in the statistical mechanics of simple two-dimensinal models. They can be very effective way of obtaining series expansions of unsolved models, and of calculating the order parameters of solved ones. Here we review these features and discuss the reason why the method fails to give the order parameter of the chiral Potts model.Comment: 18 pages, 4 figures, for Proceedings of Conference on Symmetries and Integrability of Difference Equations. (SIDE VII), Melbourne, July 200

Transfer matrix functional relations for the generalized tau_2(t_q) model

The $N$-state chiral Potts model in lattice statistical mechanics can be obtained as a ``descendant'' of the six-vertex model, via an intermediate ``$Q$'' or ``$\tau_2 (t_q)$'' model. Here we generalize this to obtain a column-inhomogeneous $\tau_2 (t_q)$ model, and derive the functional relations satisfied by its row-to-row transfer matrix. We do {\em not} need the usual chiral Potts relations between the $N$th powers of the rapidity parameters $a_p, b_p, c_p, d_p$ of each column. This enables us to readily consider the case of fixed-spin boundary conditions on the left and right-most columns. We thereby re-derive the simple direct product structure of the transfer matrix eigenvalues of this model, which is closely related to the superintegrable chiral Potts model with fixed-spin boundary conditions.Comment: 21 pages, 5 figure

The bulk, surface and corner free energies of the square lattice Ising model

We use Kaufman's spinor method to calculate the bulk, surface and corner free energies $f_b, f_s, f_s', f_c$ of the anisotropic square lattice zero-field Ising model for the ordered ferromagnetic case. For $f_b, f_s, f'_s$ our results of course agree with the early work of Onsager, McCoy and Wu. We also find agreement with the conjectures made by Vernier and Jacobsen (VJ) for the isotropic case. We note that the corner free energy $f_c$ depends only on the elliptic modulus $k$ that enters the working, and not on the argument $v$, which means that VJ's conjecture applies for the full anisotropic model. The only aspect of this paper that is new is the actual derivation of $f_c$, but by reporting all four free energies together we can see interesting structures linking them.Comment: 43 pages, 2 figures, paper amended to acknowledge previous wor

The τ2\tau_2 model and parafermions

Paul Fendley has recently found a "parafermionic" way to diagonalise a simple solvable hamiltonian associated with the chiral Potts model. Here we indicate how this method generalizes to the $\tau_2$ model with open boundaries and make some comments.Comment: 14 pages, 1 figur

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

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.Comment: 16 pages, no figures; revised 11 Jan 2010 to correct citations and to include reference to subsequent wor