Free energy of the three-state τ2(tq)\tau_2(t_q) model as a product of elliptic functions

{We show that the free energy of the three-state $\tau_2(t_q)$ model can be expressed as products of Jacobi elliptic functions, the arguments being those of an hyperelliptic parametrization of the associated chiral Potts model. This is the first application of such a parametrization to the $N$-state chiral Potts free energy problem for $N > 2$.Comment: 20 pages, 3 figure

Some exact results for the three-layer Zamolodchikov model

In this paper we continue the study of the three-layer Zamolodchikov model started in our previous works. We analyse numerically the solutions to the Bethe ansatz equations. We consider two regimes I and II which differ by the signs of the spherical sides (a1,a2,a3)->(-a1,-a2,-a3). We accept the two-line hypothesis for the regime I and the one-line hypothesis for the regime II. In the thermodynamic limit we derive integral equations for distribution densities and solve them exactly. We calculate the partition function for the three-layer Zamolodchikov model and check a compatibility of this result with the functional relations. We also do some numerical checkings of our results.Comment: LaTeX, 27 pages, 9 figure

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

The Large N Limits of the Chiral Potts Model

In this paper we study the large-N limits of the integrable N-state chiral Potts model. Three chiral solutions of the star-triangle equations are derived, with states taken from all integers, or from a finite or infinite real interval. These solutions are expected to be chiral-field lattice deformations of parafermionic conformal field theories. A new two-sided hypergeometric identity is derived as a corollary.Comment: 41 pages, 3 figures, LaTeX 2E file, using elsart.cls and psbox.tex (version 1.31 provided), [email protected]

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

How tropical Pacific surface cooling contributed to accelerated sea ice melt from 2007 to 2012 as ice is thinned by anthropogenic forcing

Over the past 40 years the Arctic sea ice minimum in September has declined. The period between 2007 and 2012 showed accelerated melt contributed to the record minima of 2007 and 2012. Here, observational and model evidence shows that the changes in summer sea ice since the 2000s reflects a continuous anthropogenically forced melting masked by interdecadal variability of Arctic atmospheric circulation. This variation is partially driven by teleconnections originating from sea surface temperature (SST) changes in the east-central tropical Pacific via a Rossby wave train propagating into the Arctic (hereafter referred to as the “Pacific-Arctic teleconnection (PARC)”), which represents the leading internal mode connecting the pole to lower latitudes. This mode has contributed to accelerated warming and Arctic sea ice loss from 2007 to 2012, followed by slower declines in recent years, resulting in the appearance of a slowdown over the past 11 years. A pacemaker model simulation, in which we specify observed SST in the tropical eastern Pacific, demonstrates a physically plausible mechanism for the PARC mode. However, the model-based PARC mechanism is considerably weaker and only partially accounts for the observed acceleration of sea ice loss from 2007 to 2012. We also explore features of large-scale circulation patterns associated with extreme melting periods in a long (1800-yr) CESM preindustrial simulation. These results further support the role of remote SST forcing originating from the tropical Pacific in exciting significant warm episodes in the Arctic. However, further research is needed to identify the reasons for model limitations in reproducing the observed PARC mode featuring a Cold Pacific - Warm Arctic connection