Critical dimensionalities of phase transitions on fractals

Abstract

Several arguments are given leading to the sufficient and necessary condition for spontaneous symmetry breaking at a finite temperature on fractals, which is d̃≥2 for discrete symmetry and d≥${\mathit{d}}_{\mathit{w}}$+1 for continuous symmetry, where d̃, d, and ${\mathit{d}}_{\mathit{w}}$ are, respectively, the spectral dimensionality, fractal dimensionality, and dimensionality of the random walk of this structure. In addition, phase transitions can always occur at ${\mathit{T}}_{\mathit{c}}$>0 on infinitely ramified lattices. Since d̃<2 for fractals usually studied, ${\mathit{T}}_{\mathit{c}}$ was always found to be 0 on finitely ramified fractals. d̃≥2 can be satisfied by a bifractal, a Cartesian product of two fractals, hence ${\mathit{T}}_{\mathit{c}}$>0 is expected. A Peierls-Griffiths proof is given for an Ising model on an example of bifractals, the periodic Koch lattice with d̃=2, showing that ${\mathit{T}}_{\mathit{c}}$ is indeed finite. A unified picture concerning both fractal and Euclidean lattices is thus obtained.