15,188 research outputs found

### 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

### 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

### 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

### 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

### The Complex of Solutions of the Nested Bethe Ansatz. The A_2 Spin Chain

The full set of polynomial solutions of the nested Bethe Ansatz is
constructed for the case of A_2 rational spin chain. The structure and
properties of these associated solutions are more various then in the case of
usual XXX (A_1) spin chain but their role is similar

### Gaudin Hypothesis for the XYZ Spin Chain

The XYZ spin chain is considered in the framework of the generalized
algebraic Bethe ansatz developed by Takhtajan and Faddeev. The sum of norms of
the Bethe vectors is computed and expressed in the form of a Jacobian. This
result corresponds to the Gaudin hypothesis for the XYZ spin chain.Comment: 12 pages, LaTeX2e (+ amssymb, amsthm); to appear in J. Phys.

### The six and eight-vertex models revisited

Elliott Lieb's ice-type models opened up the whole field of solvable models
in statistical mechanics. Here we discuss the ``commuting transfer matrix'' $T,
Q$ equations for these models, writing them in a more explicit and transparent
notation that we believe offers new insights. The approach manifests the
relationship between the six-vertex and chiral Potts models, and between the
eight-vertex and Kashiwara-Miwa models.Comment: 30 pages, 6 figure

### 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

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