We find a decoupling limit of planar ≤ 4 super Yang-Mills (SYM) on × S³ in which it becomes equivalent to the ferromagnetic XXX_1/2 Heisenberg spin chain in an external magnetic field. The decoupling limit generalizes the one found in ref. [4] corresponding to the case with zero magnetic field. The presence of the magnetic field is seen to break the degeneracy of the vacuum sector and it has a non-trivial effect on the low energy spectrum. We find a general connection between the Hagedorn temperature of planar ≤ 4 SYM on × S³ in the decoupling limit and the thermodynamics of the Heisenberg chain. This is used to study the Hagedorn temperature for small and large value of the effective coupling. We consider the dual decoupling limit of type IIB strings on AdS₅ × S ⁵. We find a Penrose limit compatible with the decoupling limit that gives a magnetic pp-wave background. The breaking of the symmetry by the magnetic field on the gauge theory side is seen to have a geometric counterpart in the derivation of the Penrose limit. We take the decoupling limit of the pp-wave spectrum and succesfully match the resulting spectrum to the low energy spectrum on the gauge theory side. This enables us to match the Hagedorn temperature of the pp-wave to the Hagedorn temperature of the gauge theory for large effective coupling. This generalizes the results of ref. [5] to the case of non-zero magnetic field.