Statistical mechanics of a fractal lattice: Renormalization-group analysis of the Sierpinski gasket

Exact renormalization-group recursion relations are constructed for several thermodynamic properties of the Sierpinski gasket (SG). A model with one, two, and three spin interactions is studied. The interactions are restricted to be explicitly part of the fractal geometry. The specific heat, paramagnetic susceptibility, and near-neighbor correlation function are calculated exactly both near and away from criticality which occurs at zero temperature (infinite near-neighbor coupling). With one exception, our results agree with the asymptotic critical properties of the SG as discussed by Gefen et al. As a finitely ramified structure, the critical exponents of the SG are essentially one dimensional. The nonuniversal properties of the SG, however, are markedly different from those in one dimension. The scale-invariant, fractal structure of the SG leads to strong critical fluctuations and, therefore, to a rather wide critical region. We find a logarithmic correction to the leading singularity in the correlation length which is necessary to account for the approach to the critical region. The specific heat of the SG is qualitatively different from that in one dimension. A continuous line of fixed points at zero temperature is also found. This is due to a marginal operator associated with a competition between the two (odd-spin) symmetry-breaking fields. In the absence of a two-spin interaction, the SG becomes frustrated when the two odd-spin interactions compete directly. The phase diagram in this limit is also calculated.