The Superintegrable Chiral Potts Model On Lattices Of Small Width

Mathematics Research Report MRR95-030

The Superintegrable Chiral Potts Model On Lattices Of Small Width

R.J. Baxter

Abstract: The order parameter of the chiral Potts model can be defined as the expectation value of an operator (Z_r)^j acting on a spin in column r. We expect it to be rapidity-independent and the same for the general model as for the superintegrable model with fixed-spin boundary conditions. For lattices of width L=1,2 or 3, we obtain explicit representations of the superintegrable chiral Potts Hamiltonian in the minimal invariant sub-space needed for these boundary conditions. We also obtain the matrix elements of (Z_r)^j between the basis vectors. From these we obtain small-L approximations for the order parameter. These approximations would become exact if we could take the large-L limit. As it is we merely repeat the verification of the 1989 conjecture of Albertini et al to order k'^2L-1 in a low-temperature expansion in the Hamiltonian coupling parameter k'. We embarked on this calculation in the hope that we would see some ``Ising-like'' pattern that would enable us to handle the L$\ra \infty$ limit. This we have so far failed to do, but present the results in the hope that they may yet prove useful.

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