TY - JOUR

T1 - The ice model and the eight-vertex model on the two-dimensional Sierpinski gasket

AU - Chang, Shu Chiuan

AU - Chen, Lung Chi

AU - Lee, Hsin Yun

PY - 2013/4/15

Y1 - 2013/4/15

N2 - 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 lim v→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)= 22(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.

AB - 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 lim v→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)= 22(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.

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U2 - 10.1016/j.physa.2013.01.005

DO - 10.1016/j.physa.2013.01.005

M3 - Article

AN - SCOPUS:84873724018

SN - 0378-4371

VL - 392

SP - 1776

EP - 1787

JO - Physica A: Statistical Mechanics and its Applications

JF - Physica A: Statistical Mechanics and its Applications

IS - 8

ER -