2012/03/06 by Shu-Chiuan Chang, Lung-Chi Chen, Hsin-Yun Lee · 1 citation
Physics and Astronomy · #cond-mat.stat-mech
paper · pdf · doi:10.1016/j.physa.2013.01.005
published as Physica A 392, 1776-1787 (2013) · 20 pages, 6 figures, 2 tables
arxiv created 2012/03/06 · arxiv updated 2013/02/19
We present the numbers of ice model and eight-vertex model configurations (with Boltzmann factors equal to one), I(n) and E(n) respectively, on the two-dimensional Sierpinski gasket SG(n) at stage n. For the eight-vertex model, the number of configurations is E(n)=23(3n+1)/2 and the entropy per site, defined as limv → ∞ ln E(n)/v where v is the number of vertices on SG(n), is exactly equal to ln 2. For the ice model, the upper and lower bounds for the entropy per site limv → ∞ ln I(n)/v 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 accurate. The corresponding result of ice model on the generalized two-dimensional Sierpinski gasket SGb(n) with b=3 is also obtained. For the generalized vertex model on SG3(n), the number of configurations is 2(8 × 6n +7)/5 and the entropy per site is equal to \frac87 ln 2. The general upper and lower bounds for the entropy per site for arbitrary b are conjectured.