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

• Published in: Journal of statistical physics 2023-10, Vol.190 (11), Article 170
• Main Authors: Lima, Paulo C., Mariani, Riccardo, Procacci, Aldo, Scoppola, Benedetto
• Format: Article
• Language: English
• Subjects: Advertising executives; Analysis; Ferromagnetism; Magnetization; Mathematical and Computational Physics; Physical Chemistry; Physics; Physics and Astronomy; Quantum Physics; Statistical Physics and Dynamical Systems; Theoretical
• ISSN: 1572-9613; 0022-4715; 1572-9613
• DOI: 10.1007/s10955-023-03181-9

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-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 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.