Studies of the Migdal-Kadanoff renormalization group for Ising systems

We study the critical temperatures obtained from the Migdal-Kadanoff (MK) renormalization transformation scheme for various ferromagnetic and frustrated Ising systems. We point out various problems in previous arguments and methods for obtaining the ordering temperatures for spin glasses using the MK approach, mostly due to the inadequate consideration of the size b of the renormalization cell used. We trace the apparent MK critical temperatures T* as a function of b for the different systems. We find empirically that the optimal value of b = b*, which gives T*(b) equal to the ordering temperature, is independent of the details of the set of interactions for a given family of systems at fixed dimension. This allows us to estimate the ordering temperatures for various types of interactions. Our results indicate that frustated system always correspond to higher values of b* than ferromagnetic ones of the same dimension. Finally, we argue that larger values of b* correspond to a larger amount of information that must be incorporated at the level of the basic unit to correctly describe more complex ground states.