Without more information (like, magnitude of difference, and which initial states considered) it's hard to say anything specific, but there is at least the possibility that you end up in either the ferromagnetic or anti-ferromagnetic states (basically, if your edges have the same or opposite signs of the initial polarizations). Typically the anti-ferromagnetic state has a lower total energy and is thus the true ground state, but you can't reach the anti-ferromagnetic state from the ferromagnetic state without crossing an energy barrier, so if your initial configuration is ferromagnetic, you will converge to such a state, and find the correct lowest energy within the ferromagnetic subspace.
