Serge Galam, André C. R. Martins
A novel intermediate "dis/order" phase is uncovered between the usual ordered and disordered phases for the classical Ising ferromagnetic model on a square lattice. Applying the Galam Unifying Frame (GUF), set to investigate the dynamics of two-state variable systems within the frame of opinion dynamics, combined to either Metropolis or Glauber scheme, reveals a twofold order-disorder Ising transition. The transition is no longer at once from the ordered state to the disordered state but instead goes continuously through a phase for which two equilibrium states exist, each one being associated to a different basin of attraction of initial conditions. Their respective sizes are a function of the temperature. The one to the disordered phase starts from zero size at a first critical temperature $T_{c1}$ to embody the total landscape of initial conditions at a second critical temperature $T_{c2}$. The second one corresponds to the usual critical temperature. In $J/k_B$ units the Metropolis scheme yields $T_{c1}\approx 1.59$ and $T_{c2}\approx 2.11$ while Glauber gives $T_{c1}\approx 1.58$ and $T_{c2}\approx 2.17$. Comparing to the Onsager result $T_c\approx 2.27$ shows a drastic and significant improvement in the accuracy of the value of $T_c$ obtained using the simple GUF analytical scheme with a discrepancy reduced to around 4% for Glauber. This improvement is a solid result while the appearance of the intermediate "dis/order" phase still needs to be confirmed eventually using exact methods.
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http://arxiv.org/abs/1303.2957
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