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 SK solution $Q^{({rm RS})}_{ab} = delta_{ab} + q(1-delta_{ab})$ is investigated. In particular, when the expansion is truncated to fourth order in. $Q_{ab} - Q^{({rm RS})}_{ab}$. The Full Replica Symmetry Broken (FRSB) solution is explicitly found but it turns out to exist only in the range of temperature $0.549...leq Tleq T_c=1$, not including T=0. On the other hand an expansion around the paramagnetic solution $Q^{({rm PM})}_{ab} = delta_{ab}$ up to fourth order yields a FRSB solution that exists in a limited temperature range $0.915...leq T leq T_c=1$.