Log-periodic oscillations for a uniform spin model on a fractal

The model of Blume-Capel on the Sierpinski gasket is investigated within the method of transfer matrices, where the thermodynamic functions are obtained after the numerical iteration of a set of discrete maps. The analysis of the T=0 transition shows that, for antiferromagnetic coupling and a finite interval of self-energy coefficient, the correlation length diverges as exp(J(eff)/T), with superimposed log-periodic oscillations in terms of the reduced temperature t=exp(-|J(eff)|/T). Both the period of oscillations and the effective interaction J(eff) depend on the strength of the actual coupling constants. In the antiferromagnetic regime, residual entropy is found for three different values of the self-energy parameter. The variation of this parameter leads, in the case of ferromagnetic coupling, to a more complex behavior for the correlation length than the already known exp[exp(J(eff)/T)] dependence observed for the Ising and Potts models.