Frustrated bond in an antiferromagnetic Ising ring ($S_j{S}_{j+1}=+1$)

A neighboring parallel pair in an antiferromagnetic Ising ring. It costs $2J$ relative to a satisfied antiparallel pair. If there are $m$ such pairs, the energy is $-JN+2Jm$, and the periodic boundary conditions force $N-m$ to be even because ${\prod }_j{S}_j{S}_{j+1}=1$. Thus an odd ring has $2N$ ground states, obtained by locating one such pair and choosing one overall Ising spin orientation. The even ring has two ground states with no such pair. This definition uses the local preferred ordering; a parallel pair is a defect in staggered order rather than a wall between uniformly magnetized domains.