Renormalization Group on hierarchical lattices in finite dimensional disordered Ising and Blume-Emery-Griffiths Models

Renormalization group on hierarchical lattices is often considered a valuable tool to understand the critical behavior of more complicated statistical mechanical models. In presence of quenched disorder, however, in many model cases predictions obtained with the Migdal-Kadanoff bond removal approach fail to quantitatively and qualitatively reproduce critical properties obtained in the mean-field approximation or by numerical simulations in finite dimensions. In order to critically review this limitation we analyze the behavior of Ising and Blume-Emery-Griffiths models on more complicated hierarchical lattices. We find that, apart from some exceptions, the different behavior appears not only limited to Midgal-Kadanoff-like cells but is associated right to the hierarchization of Bravais lattices in small cells also when in-cell loops are considered.