With the help of the renormalization method we have calculated the low-temperature expansion series of free energy for the Ising models that are based on a fractal structure and with zero critical temperature. The results show that there exists a critical value of vd¯ such that, when vd¯ is larger than the critical value, scaling theory fails and the hyperscaling law is broken. We also found that the Ising models with zero critical temperature belong to two series of universality classes, respectively, one of which mainly corresponds to the Ising models based on the lattices generated in a site self-similar manner, which are named site-hierarchical lattices. Another series involves all Ising models that are based on other fractal structures. These two universality classes have different critical conditions of scaling and approach different thermal exponents α when vd¯ becomes larger than the critical value.

• Received 12 March 1990

DOI:https://doi.org/10.1103/PhysRevA.42.3259