The Blume-Emery-Griffiths model at the FAD and AD interfaces

Abstract

We analyse the Blume-Emery-Griffiths (BEG) model on the lattice $\Zd$ at the ferromagnetic-antiquadrupolar-disordered (FAD) and antiquadrupolar-disordered (AD) interfaces of parameters. In our analysis of the FAD interface we introduce a Gibbs sampler of the ground states at zero temperature, and we exploit it in two different ways: first, we perform via perfect sampling an empirical evaluation of the spontaneous magnetization at zero temperature, finding a non-zero value in $d=3$ and a vanishing value in $d=2$. Second, using a careful coupling with the Bernoulli site percolation model in $d=2$, we prove rigorously that imposing $+$ boundary conditions, the magnetization in the center of a square box tends to zero in the thermodynamical limit and the two-point correlations decay exponentially. Also, using again a coupling argument, we show that the infinite volume Gibbs measure of the zero-temperature BEG exists and it is unique. In our analysis of the AD interface we restrict ourselves to $d=2$ and, by comparing the BEG model with a Bernoulli site percolation in a matching graph of ${\mathbb{Z}}^2$, we get a condition for the vanishing of the infinite volume limit magnetization improving, for low temperatures, earlier results obtained via expansion techniques.

References in corpus (7)

• The repulsive lattice gas, the independent-set polynomial, and the Lovász local lemma
• Square lattice site percolation at increasing ranges of neighbor interactions
• The 3-State Square-Lattice Potts Antiferromagnet at Zero Temperature
• Blume-Emery-Griffiths model on the square lattice with repulsive biquadratic coupling
• Potts model on infinite graphs and the limit of chromatic polynomials
• Absolute convergence of the free energy of the BEG model in the disordered region for all temperatures
• Uniqueness of the Gibbs state of the BEG model in the disordered region of parameters