Ice model and eight-vertex model on the two-dimensional Sierpinski gasket

Abstract

We present the numbers of ice model configurations (with Boltzmann factors equal to one) I(n)I(n) on the two-dimensional Sierpinski gasket SG(n)SG(n) at stage nn. The upper and lower bounds for the entropy per site, defined as limv→∞lnI(n)/vlimv→∞lnI(n)/v, where vv is the number of vertices on SG(n)SG(n), are derived in terms of the results at a certain stage. As the difference between these bounds converges quickly to zero as the calculated stage increases, the numerical value of the entropy can be evaluated with more than a hundred significant figures accuracy. The corresponding result of the ice model on the generalized two-dimensional Sierpinski gasket SGb(n)SGb(n) with b=3b=3 is also obtained, and the general upper and lower bounds for the entropy per site for arbitrary bb are conjectured. We also consider the number of eight-vertex model configurations on SG(n)SG(n) and the number of generalized vertex models Eb(n)Eb(n) on SGb(n)SGb(n), and obtain exactly Eb(n)=2{2(b+1)[b(b+1)/2]n+b+4}/(b+2)Eb(n)=2{2(b+1)[b(b+1)/2]n+b+4}/(b+2). It follows that the entropy per site is View the MathML sourcelimv→∞lnEb(n)/v=2(b+1)b+4ln2.