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

TL;DR

This paper critically examines the effectiveness of renormalization group methods on hierarchical lattices for disordered models, revealing limitations beyond simple Migdal-Kadanoff approximations and highlighting the impact of lattice hierarchization.

Contribution

The study extends the analysis of renormalization group approaches to more complex hierarchical lattices for disordered models, identifying limitations related to lattice hierarchization.

Findings

Limitations of Migdal-Kadanoff approach in disordered models

Hierarchization of lattices affects critical property predictions

Complex hierarchical lattices reveal behavior not captured by simpler 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.