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arXiv:1402.5820 · Physical Review E

Small-cluster renormalization group in Ising and Blume-Emery-Griffiths models with ferromagnetic, antiferromagnetic, and quenched disordered magnetic interactions

Fabrizio Antenucci, A. Crisanti, Luca Leuzzi

Published Jul 14, 2014• 4 citations
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Abstract

The Ising and Blume-Emery-Griffiths (BEG) models' critical behavior is analyzed in two dimensions and three dimensions by means of a renormalization group scheme on small clusters made of a few lattice cells. Different kinds of cells are proposed for both ordered and disordered model cases. In particular, cells preserving a possible antiferromagnetic ordering under renormalization allow for the determination of the Néel critical point and its scaling indices. These also provide more reliable estimates of the Curie fixed point than those obtained using cells preserving only the ferromagnetic ordering. In all studied dimensions, the present procedure does not yield a strong-disorder critical point corresponding to the transition to the spin-glass phase. This limitation is thoroughly analyzed and motivated.

DOI
10.1103/physreve.90.012112

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The Ising and Blume-Emery-Griffiths (BEG) models' critical behavior is analyzed in two dimensions and three dimensions by means of a renormalization group scheme on small clusters made of a few lattice cells. Different kinds of cells are proposed for both ordered and disordered model cases. In particular, cells preserving a possible antiferromagnetic ordering under renormalization allow for the determination of the Néel critical point and its scaling indices. These also provide more reliable estimates of the Curie fixed point than those obtained using cells preserving only the ferromagnetic ordering. In all studied dimensions, the present procedure does not yield a strong-disorder critical point corresponding to the transition to the spin-glass phase. This limitation is thoroughly analyzed and motivated.
A prefaced renormalization-group study indicates that all three features of the Blume-Emery-Griffiths model with repulsive biquadratic interaction, namely (1) a critical-end-point structure occurring inside the ferromagnetic phase, (2) disordered-ferromagnetic-disordered reentrance, and (3) a ferrimagnetic phase sandwiched between the ferromagnetic and antiquadrupolar phases, survive fluctuations in three dimensions but not in two dimensions. The renormalization-group mechanism for a critical-end-point structure inside the ordered phase is shown to be the same as that originally found for a critical-end-point structure inside the disordered phase, namely a distinct hybrid fixed point, contrary to a previous claim.
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The Ising model with a quenched random distribution of ferromagnetic and antiferromagnetic interactions has been investigated by tracking the probability distribution of interactions under rescaling. As the initial probability of a ferromagnetic interaction is increased, the phase diagram in three dimensions shows three ordered phases: antiferromagnetic, spin glass and ferromagnetic. We find that, within the spin-glass region, effective interactions between pairs of spins at successive length scales occur in a chaotic sequence, while the average magnitudes of the ferromagnetic and the antiferromagnetic interactions both flow to infinity under iteration. Our calculated spin-glass multicritical points are on the Nishimori line in cases of both bimodal (±J) and diluted (trimodal) distributions, and these points flow under rescaling to a common fixed distribution with a specific heat exponent α of approximately −3.7.
Abstract We use the Monte Carlo simulation technique to study the critical behavior of a three-state spin model, with bilinear and biquadratic nearest-neighbor interactions, known as the Blume–Emery–Griffiths model (BEG), in a square lattice. In order to characterize this model, we study the phase diagram, in which we identify three different phases: ferromagnetic, paramagnetic and quadrupolar, the later with one sublattice filled with spins and the other with vacancies. We perform our studies by using two algorithms: Metropolis update (MU) and Wang–Landau (WL). The critical scaling behavior of the model is complementary studied by applying results obtained by using both algorithms, while tricritical points and the tricritical scaling behavior is analyzed by means of WL measuring the joint density of states and using the method of field mixing in conjunction with finite-size scaling. Furthermore, motivated by the decoupling between spins observed within the quadrupolar phase, we further generalize the BEG model in order to study the behavior of the system by adding a next nearest neighbour (NNN) interaction between spins. We found that by increasing the strength of the (ferromagnetic) NNN interaction, a new ferromagnetic phase takes over that contains both quadrupolar and ferromagnetic order.
We investigate the three-dimensional semi-infinite Blume-Emery-Griffiths (BEG) model using the Migdal-Kadanoff renormalization-group method. The parameter space is six dimensional. We find 69 fixed points describing a large variety of critical behaviors and discuss various semi-infinite models which are special cases of the BEG model such as the Blume-Capel model. For this last model we determine the various generic types of phase diagrams according to the values of the interactions on the surface and in the bulk of the system.