Sherrington-Kirkpatrick model near T=T_c: expanding around the Replica Symmetric Solution

An expansion for the free energy functional of the Sherrington–Kirkpatrick (SK) model, around the replica symmetric (RS) SK solution Q(RS)ab = δab + q(1 − δab) is investigated. In particular, when the expansion is truncated to the fourth order in Qab − Q(RS)ab the full replica symmetry broken (FRSB) solution is explicitly found but it turns out to exist only in the range of temperature 0.549... ≤ T ≤ Tc = 1, not including T = 0. On the other hand, an expansion around the paramagnetic solution Q(PM)ab = δab, up to the fourth order, yields a FRSB solution that exists in a limited temperature range 0.915... ≤ T ≤ Tc = 1.