  We study the strong coupling behaviour of fixed length single trace operators
in the scalar SU(2) sector of ${\cal N} = 4$ SYM. We assume the recently
proposed connection with a twisted half-filled Hubbard model. By explicit
direct diagonalization of operators with length $L=4, 6, 8$ we study the full
{\em perturbative multiplet} of those lattice states which have a clear
correspondence with gauge theory composite operators. For this multiplet, we
follow the weak-strong coupling flow to free fermion states and identify in
particular the precise asymptotic fermion configuration. Next, we analyze the
Lieb-Wu equations of the twisted Hubbard model. For the antiferromagnetic state
we derive its strong coupling expansion working at $L$ up to 32. We also study
the lightest state in the perturbative multiplet. This state is non trivial
since involves complex solutions of the Lieb-Wu equations. It is particularly
interesting for $AdS_5\times S^5$ duality since it is dual to the folded string
semiclassical solution in the thermodynamical limit. We are able to perform the
full analysis and compute the next-to-next-to leading terms in the strong
coupling expansion for the non trivial lengths L=12 and L=20. A general formula
is proposed for the NLO expansion for any $L=4(2k+1)$, $k\in\mathbb{N}$.
