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. $Q_{ab}-Q_{ab}^{(\mathrm{R}\mathrm{S})}$. The Full Replica Symmetry Broken (FRSB) solution is explicitly found but it turns out to exist only in the range of temperature $\mathrm{0.549...}\le T\le T_c=1$, not including T=0. On the other hand an expansion around the paramagnetic solution $Q_{ab}^{(\mathrm{P}\mathrm{M})}={\delta }_{ab}$ up to fourth order yields a FRSB solution that exists in a limited temperature range $\mathrm{0.915...}\le T\le T_c=1$.