Fig. 8: Ground state Ising spin configurations.

a State S0 where each neighboring spin alternates, and b State S1 where there are two positions, opposite on the ring, for which neighboring Ising spins are the same. For J < Jcrit, the neighboring couplings dominate, and S0 is the ground state, whereas for J > Jcrit, the cross-ring couplings exert a greater influence and S1 is the ground state.