On the Ashkin-Teller model and Tutte-Whitney functions

The partition functions of the Ising and Potts models in statistical mechanics are well known to be partial evaluations of the Tutte-Whitney polynomial of the appropriate graph. The Ashkin-Teller model generalizes the Ising model and the four-state Potts model, and has been extensively studied since its introduction in 1943. However, its partition function (even in the symmetric case) is not a partial evaluation of the Tutte-Whitney polynomial. In this paper, we show that the symmetric Ashkin-Teller partition function can be obtained from a generalized Tutte-Whitney function which is intermediate in a precise sense between the usual Tutte-Whitney polynomial of the graph and that of its dual.