The Blume-Emery-Griffiths Model on the FAD Point and on the AD Line

We analyse the Blume-Emery-Griffiths (BEG) model on the lattice Z(d) on the ferromagnetic-antiquadrupolar-disordered (FAD) point and on the antiquadrupolar-disordered (AD) line. In our analysis on the FAD point, 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(z)ero 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 under 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 there exists a unique zero-temperature infinite-volume Gibbs measure for the BEG. In our analysis of the AD line we restrict ourselves to d = 2 and, by comparing the BEG model with a Bernoulli site percolation in a matching graph of 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.

Automated summary

In this article, the Blume-Emery-Griffiths (BEG) model is analyzed in different states, and its properties and behaviors are studied. In the analysis of the ferromagnetic-antiquadrupolar-disordered (FAD) point, a Gibbs sampler of the ground states at zero temperature is introduced for empirical evaluation of the spontaneous magnetization. The results show different values in different dimensions. Moreover, a coupling with the Bernoulli site percolation model is used to prove the vanishing of magnetization in the center of a square box under certain boundary conditions in the thermodynamical limit, and exponential decay of two-point correlations. In the analysis of the antiquadrupolar-disordered (AD) line, a condition for the vanishing of the infinite-volume limit magnetization is obtained by comparing the BEG model with a Bernoulli site percolation, improving previous results obtained via expansion techniques.