On the role of spectral dimension in determining phase transition

We have studied the phase transition of the Ising model on a family of fractals called Nice trees whose spectral dimension d s can take values greater than 2. The phase transition is shown to be the trivial zero-temperature one through exactly solving the free energy and the spontaneous magnetization of the system, and is different from that on Cayley trees. The result is independent of d s of the structure and hence provides an example of trivial phase transition at d s >or=2, which does not agree with the argument of Yu and Gong (1994). It suggests that the role of d s in determining the phase transition may be complex.