Exact results of AT model on a special kind of Sierpinski carpets

Exact results are obtained for the Ashkin-Teller model on a special Sierpinski carpet (SC). We calculate the partition function and correlation function with the transformation matrix method. Thus the asymptotic behaviour of the correlation function is investigated in the limit of long distance. The results show that no phase transition occurs at any finite temperature for the finite R (the order of ramification) fractal, which is consistent with the conclusion of the Ising and Potts model.