the ferromagnetic state, we expect that the separation between majority-spin and minority-spin extended state levels will increase rapidly so that the $\nu=1$ integer quantum Hall plateau will broaden and become observable. At still larger $\gamma$, we find a transition to a state with the maximum spin polarization allowed by the Pauli exclusion principle. At $\nu \le 1$, this is full spin-polarization. In these states, marked ‘SG’ for spin-gap in Fig. \[phasediag\], the differential spin-susceptibility vanishes. For realistic disorder models, it seems likely that in the thermodynamic limit there will always be rare high-disorder regions in the sample which prevent maximal spin-polarization from being achieved. For this reason, the phase transition we find in our finite systems likely indicates a crossover from large to small differential spin-susceptibility in macroscopic systems; we have therefore marked this transition by a dashed line. Finally at the largest values of $\gamma$ (weakest disorder) the physics for $\nu$ near $1$ is dominated by Skyrmion quasiparticles which emerge from the $\nu=1$ ferromagnetic vacuum. In this regime, the system develops spontaneous spin-polarization in the plane perpendicular to the direction of the Zeeman field. In Fig. \[phasediag\] we have labelled this regime NCF for non-collinear ferromagnet.

This phase diagram 