Crossover from a fractal lattice to a Euclidean lattice for the thermodynamic properties of a triplet-interaction Ising model.

Abstract

We study how thermodynamic properties of the triplet-interaction Ising model on a family of Sierpinski-type gasket fractals cross over to that of the Ising model on a triangular lattice with three-spin interaction in half of the triangular faces. By using a spin-variable transformation, the exact free energy ${\mathit{f}}_{\mathit{t}}$ for the triangular model and ${\mathit{f}}_{\mathit{s}}$ for the fractals are obtained in closed form and found to be analytic in temperature. The free energy ${\mathit{f}}_{\mathit{s}}$ varies smoothly with the parameter b (which labels each member of the fractal family), and for large b the difference between ${\mathit{f}}_{\mathit{t}}$ and ${\mathit{f}}_{\mathit{s}}$ is asymptotically directly proportional to 1/b. For fractal lattices with b=${2}^{\mathit{m}}$ (m=1,2,3,. . .), the crossover behavior of the critical exponents is also discussed by using a renormalization-group approach. In the meantime we find that the correlation-length exponent \ensuremath{\nu}=ln2/ln3, which is independent of the parameter b and hence the fractal dimension ${\mathit{d}}_{\mathit{f}}$ and is different from \ensuremath{\nu}=\ensuremath{\infty} for the two-spin-interaction Ising model on the fractal lattice given by Gefen et al. It shows that the universality hypothesis is violated here.