CRITICAL DIMENSIONALITIES OF PHASE-TRANSITIONS ON FRACTALS

Several arguments are given leading to the sufficient and necessary co ndition for spontaneous symmetry breaking at a finite temperature on f ractals, which is (d) over tilde greater than or equal to 2 for discre te symmetry and d > d(w) + 1 for continuous symmetry, where (d) over t ilde, d, and d(w) are, respectively, the spectral dimensionality, frac tal dimensionality, and dimensionality of the random walk of this stru cture. In addition, phase transitions can always occur at T-c > O on i nfinitely ramified lattices. Since (d) over tilde < 2 for fractals usu ally studied, T-c was always found to be O on finitely ramified fracta ls. (d) over tilde > 2 can be satisfied by a bifractal, a Cartesian pr oduct of two fractals, hence T-c > O is expected. A Peierls-Griffiths proof is given for an Ising model on an example of bifractals, the per iodic Koch lattice with (d) over tilde = 2, showing that T-c is indeed finite. A unified picture concerning both fractal and Euclidean latti ces is thus obtained.