Applying a new method to the 2-d Ising transition

Abstract

The Ising ferromagnetic model on a square lattice is revisited using the Galam Unifying Frame (GUF), set to investigate the dynamics of two-state variable systems within the frame of opinion dynamics. When combined with Metropolis dynamics, an unexpected intermediate "dis/order" phase 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 $T_{c1}$ to embody the total landscape of initial conditions at a second critical temperature $T_{c2}$ with $T_{c1}\approx 1.59$ and $T_{c2}\approx 2.11$ in $J/k_B$ units. It appears that $T_{c2}$ is close to the Onsager result $T_{c}\approx 2.27$. The transition, which is first-order like, exhibits a vertical jump to the disorder phase at $T_{c2}$, 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 single transition at $T_{c}\approx 3.09$. Accordingly, although the "dis/order" phase 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.