  The anomalous dimensions of local single trace gauge invariant operators in
N=4 supersymmetric Yang-Mills theory can be computed by diagonalizing a long
range integrable Hamiltonian by means of perturbative asymptotic Bethe ansatz.
This formalism breaks down when the number of fields of the composite operator
is smaller than the range of the Hamiltonian which coincides with the order in
perturbation theory at study. We analyze two spin chain toy models which might
shed some light on the physics behind these wrapping effects. One of them, the
Hubbard model, is known to be closely related to N=4 SYM. In this example, we
find that the knowledge of the effective spin chain description is insufficient
to reconstruct the finite size effects of the underlying electron theory. We
compute the wrapping corrections for generic states and relate them to a
Luscher like approach. The second toy models are long range integrable
Hamiltonians built from the standard algebraic Bethe anstatz formalism. This
construction is valid for any symmetry group. In particular, for non-compact
groups it exhibits an interesting relation between wrapping interactions and
transcendentality.
