Crossover from fractal lattice to Euclidean lattice for the residual entropy of an Ising antiferromagnet in maximum critical fieldHc

How thermodynamic properties of fractal objects cross over to the corresponding thermodynamic properties of nonfractal ‘‘Euclidean’’ objects is an open question of considerable recent interest. We study how the ground-state entropy σ of the Ising antiferromagnet on a family of two-dimensional fractals ‘‘crosses over’’ to the ground-state entropy of a triangular lattice. The fractal family studied is a generalization of the simple Sierpiński gasket. We find that σ varies smoothly with a parameter b (which labels each member of the fractal family) and approaches for large b the value ${\sigma }_{\mathrm{Baxter}=0.333\mathrm{}24272...}$ calculated by Baxter and Tsang for the hard-hexagon problem on the triangular lattice and confirmed by Baxter to be an exact value.