Critical phenomena on fractal lattices

Abstract

Renormalization-group techniques are applied to Ising-model spins placed on the sites of several self-similar fractal lattices. The resulting critical properties are shown to vary with the (noninteger) fractal dimensionality D, but also with several topological factors: ramification, connectivity, lacunarity, etc. For any D>~1, there exist systems with both Tc=0, and Tc>0; hence a lower critical dimensionality is not defined. The nonvanishing values of Tc and the critical exponents depend on all these factors. © 1980 The American Physical Society.