Two-dimensional Ising transition through a technique from two-state opinion-dynamics models
Physical Review E
1 - 8 p.
sociophysics, opinion dynamics, Ising model
The Ising ferromagnetic model on a square lattice is revisited using the Galam unifying frame (GUF), set to investigate two-state opinion-dynamics models. When combined with Metropolis dynamics, an unexpected intermediate “dis/order” regime is found with the coexistence of two attractors associated, respectively, to an ordered and a disordered phases. The basin of attraction of initial conditions for the disordered phase attractor starts from zero size at a first critical temperature Tc1 to embody the total landscape of initial conditions at a second critical temperature Tc2, with Tc1 ≈ 1.59 and Tc2 ≈ 2.11 in J/kB units. It appears that Tc2 is close to the Onsager result Tc ≈ 2.27. The transition, which is first-order-like, exhibits a vertical jump to the disorder phase at Tc2, reminiscent of the rather abrupt vanishing of the corresponding Onsager second-order transition. However, using Glauber dynamics combined with GUF does not yield the intermediate phase and instead the expected classical mean-field transition is recovered at Tc ≈ 3.09. Accordingly, although the “dis/order” regime produced by the GUF-Metropolis combination is not physical, it is an intriguing result to be understood. In particular the fact that Glauber and Metropolis dynamics yield so different results using GUF needs an explanation. The possibility of extending GUF to larger clusters is discussed.