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The ice model and the eight-vertex model on the two-dimensional Sierpinski gasket

Author

Listed:
  • Chang, Shu-Chiuan
  • Chen, Lung-Chi
  • Lee, Hsin-Yun

Abstract

We present the numbers of ice model configurations (with Boltzmann factors equal to one) I(n) on the two-dimensional Sierpinski gasket SG(n) at stage n. The upper and lower bounds for the entropy per site, defined as limv→∞lnI(n)/v, where v is the number of vertices on 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) with b=3 is also obtained, and the general upper and lower bounds for the entropy per site for arbitrary b are conjectured. We also consider the number of eight-vertex model configurations on SG(n) and the number of generalized vertex models Eb(n) on SGb(n), and obtain exactly Eb(n)=2{2(b+1)[b(b+1)/2]n+b+4}/(b+2). It follows that the entropy per site is limv→∞lnEb(n)/v=2(b+1)b+4ln2.

Suggested Citation

  • Chang, Shu-Chiuan & Chen, Lung-Chi & Lee, Hsin-Yun, 2013. "The ice model and the eight-vertex model on the two-dimensional Sierpinski gasket," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(8), pages 1776-1787.
    • Handle: RePEc:eee:phsmap:v:392:y:2013:i:8:p:1776-1787
    DOI: 10.1016/j.physa.2013.01.005
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