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

Abstract

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.

36 pages, 18 figures, 10 tables

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• The random field Blume-Capel model revisited
• Real Space Migdal-Kadanoff Renormalisation of Glassy Systems: Recent Results and a Critical Assessment
• Small clusters Renormalization Group in 2D and 3D Ising and BEG models with ferro, antiferro and quenched disordered magnetic interactions
• Dynamical arrest with zero complexity: the unusual behavior of the spherical Blume Emery Griffiths disordered model
• Information-Geometric Signatures from Nonextensivity in the $1$-D Blume-Capel Model
• Hierarchical Random Graphs Based on Motifs