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Critical Study of Hierarchical Lattice Renormalization Group in Magnetic Ordered and Quenched Disordered Systems: Ising and Blume–Emery–Griffiths Models

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Abstract

Renormalization group based on the Migdal–Kadanoff bond removal approach is often considered a simple and valuable tool to understand the critical behavior of complicated statistical mechanical models. In presence of quenched disorder, however, predictions obtained with the Migdal–Kadanoff bond removal approach quite often fail to quantitatively and qualitatively reproduce critical properties obtained in the mean-field approximation or by numerical simulations in finite dimensions. In an attempt to overcome this limitation we analyze the behavior of Ising and Blume–Emery–Griffiths models on more structured hierarchical lattices. We find that, apart from some exceptions, the failure is not limited to Midgal–Kadanoff cells but originates right from the hierarchization of Bravais lattices on small cells, and shows up also when in-cell loops are considered.

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Notes

  1. Previous results [47] on this lattice suffers from some inconsistency, as discussed in Ref. [48].

  2. The argument of the logarithm is always positive because the variance can either shrink or increase but does not oscillate between different RSRG steps.

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Acknowledgments

The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme FP7/2007-2013/ under REA grant agreement n 290038, NETADIS project and from the Italian MIUR under the Basic Research Investigation Fund FIRB2008 program, Grant No. RBFR08M3P4, and under the PRIN2010 program, grant code 2010HXAW77-008.

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  1. Dipartimento di Fisica, Università Sapienza, P.le Aldo Moro 2, 00185 , Rome, Italy

    F. Antenucci, A. Crisanti & L. Leuzzi

  2. IPCF-CNR, UOS Roma Kerberos, P.le Aldo Moro 2, 00185 , Rome, Italy

    F. Antenucci & L. Leuzzi

  3. ISC-CNR, UOS Sapienza, P.le Aldo Moro 2, 00185 , Rome, Italy

    A. Crisanti

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  1. F. Antenucci
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Antenucci, F., Crisanti, A. & Leuzzi, L. Critical Study of Hierarchical Lattice Renormalization Group in Magnetic Ordered and Quenched Disordered Systems: Ising and Blume–Emery–Griffiths Models. J Stat Phys 155, 909–931 (2014). https://doi.org/10.1007/s10955-014-0977-z

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