Elliptic Parametrization of the Three-State Chiral Potts Model

2002

MathPhys Odyssey 2001

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The normalization factor in the star-triangle relation can be evaluated in a simple form by taking determinants. If we combine this with the rotation symmetries, then we can show that a certain simple quantity I has to be independent of the rapidities. In this sense it is an invariant. We evaluate it for several particular models and find it is one for self-dual models, and is related to the modulus k (or k ′ ) for the Ising, Kashiwara-Miwa and chiral Potts models. It may be possible to absorb S(a) into W pq (a, b) and/or W pq (a, b), but to the author it seems clearer not to do so. † There are other solvable planar models, derived from the three-dimensional Zamolodchikov model, which are asymmetric, but these satisfy the "star-star" relation, rather than the star-triangle. [2]

Section: Chiral Potts Modelmentioning

confidence: 99%

A Rapidity-Independent Parameter in the Star-Triangle Relation

2002

MathPhys Odyssey 2001

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“…These are the relations (27) of [10]. The relations (32) of [10] are also satisfied: for all /, j . They permute the three elements z p , -l/w p , -w p /z p of u p and multiply them by powers of x, the product of the elements remaining unity.…”

Section: (44)mentioning

confidence: 92%

“…One still has two related arguments (here termed z p and w p ), but some of the properties can be expressed as products of Jacobi functions, each with an argument z p or w p , or some combination thereof. A number of such results have been obtained [10], [9, pages 568-569].…”

Section: F Pq {R)mentioning

confidence: 99%

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Hyperelliptic parametrisation of the generalised order parameter of the N = 3 chiral Potts model

2006

ANZIAM J.

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It has been known for some time that the Boltzmann weights of the chiral Potts model can be parametrised in terms of hyperelliptic functions, but as yet no such parametrisation has been applied to the partition and correlation functions. Here we show that for N = 3 the function 5(/,,) that occurs in the recent calculation of the order parameters can be expressed quite simply in terms of such a parametrisation.2000 Mathematics subject classification: 82B05.

“…We use the hyperelliptic parametrisation introduced in [27,28,29]. We define parameters x, z p , w p related to one another and to…”

Section: )mentioning

confidence: 99%

The challenge of the chiral Potts model

2006

J. Phys.: Conf. Ser.

The challenge of the chiral Potts model R J BaxterCentre for Mathematics and its Applications, The Australian National University, Canberra, ACT 0200, Australia Abstract. 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.