Exclude more complicated recurrent patterns in the antiferromagnetic phase

Determine whether chains with structured antiferromagnetic coupling and negative coupling strength can realize lower-free-energy configurations that revisit states in patterns more complicated than the indefinitely alternating two-state configuration.

Background

For negative structured coupling, neighboring copies favor opposite states. The analysis identifies an indefinitely alternating configuration of two states as the lowest-energy member among several tested families, including uniform states, cycles of fixed period, and the all-different family.

The authors explicitly note that this comparison does not constitute an exhaustive proof because configurations that revisit states according to more complicated patterns have not been ruled out. Establishing that no such patterns outperform the alternating state would complete the characterization of the antiferromagnetic locked phase.

References

We have not excluded chains that revisit states in more complicated patterns, so this is a tested family rather than a proof, in contrast with the $\epsilon>0$ case where the domain classification is exhaustive.

— Locking transition in coupled disordered systems  (2609.36846 - Bunin, 29 Sep 2026) in Supplemental Material, Section “Antiferromagnetic coupling,” subsection “Other cyclic solutions”