Critical Phenomena on Fractal Lattices

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 T₂=0, and T₂>0; hence a lower critical dimensionality is not defined. The nonvanishing values of T₂ and the critical exponents depend on all these factors.