= Frustrated bond in an antiferromagnetic Ising ring
{title2=$S_jS_{j+1}=+1$}
= Frustrated bonds in an antiferromagnetic Ising ring
{synonym}
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 $\prod_jS_jS_{j+1}=1$. 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.
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