Transfer matrix functional relations for the generalized τ 2 (t q) model Abstract 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 " τ 2 (t q) " model. Here we generalize this to obtain a column-inhomogeneous τ 2 (t q) model, and derive the functional… (More)
The solvable sl(n)-chiral Potts model can be interpreted as a three-dimensional lattice model with local interactions. To within a minor modification of the boundary conditions it is an Ising type model on the body centered cubic lattice with two-and three-spin interactions. The corresponding local Boltzmann weights obey a number of simple relations,… (More)
We discuss some of the difficulties that have been mentioned in the literature in connection with the Bethe ansatz for the six-vertex model and XXZ chain, and for the eight-vertex model. In particular we discuss the " beyond the equator " , infinite momenta and exact complete string problems. We show how they can be overcome and conclude that the coordinate… (More)
We discuss the hard-hexagon and hard-square problems, as well as the corresponding problem on the honeycomb lattice. The case when the activity is unity is of interest to combinatorialists. For this case we use the corner transfer matrix method to numerically evaluate the partition function per site and density to 33 or more digits of accuracy.
We derive the order parameter of the chiral Potts model, using the method of Jimbo et al. The result agrees with previous conjectures.
We adapt our previous results for the " partition function " of the superin-tegrable chiral Potts model with open boundaries to obtain the corresponding matrix elements of e −αH , where H is the associated hamiltonian. The spontaneous magnetization M r can be expressed in terms of particular matrix elements of e −αH S r 1 e −βH , where S 1 is a diagonal… (More)
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 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… (More)
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… (More)
We consider the sum of dichromatic polynomials over non-separable rooted planar maps, an interesting special case of which is the enumeration of such maps. We present some known results and derive new ones. The general problem is equivalent to the q-state Potts model randomized over such maps: it remains an open question whether this model exhibits a phase… (More)