Nonintegrability proof for the original non-periodic Motzkin chain

Extend the nonintegrability proof for the area-weighted periodic Motzkin chain to the original non-periodic Motzkin chain, and determine whether that model has no nontrivial k-local conserved quantities in the corresponding locality range.

Background

The paper proves the absence of k-local conserved quantities for the periodic Motzkin chain, including the area-weighted family, for 3 <= k <= N/2. The original Motzkin chain differs through its boundary conditions and has a unique ground state in the setting associated with reported spontaneous U(1) symmetry breaking. A rigorous extension of the periodic-chain argument to this non-periodic model is left unresolved.

References

A rigorous extension of the proof to the original (non-periodic) Motzkin chain, the analysis of the remaining pattern $f_0=f_{\pm1}=f_{\pm2}=0$, and a general characterization of continuous symmetry breaking in frustration-free systems without a commuting order parameter remain important directions for future work.

— Absence of nontrivial local conserved quantities in a class of $U(1)$-symmetric spin-1 chains  (2608.17548 - Sengoku et al., 18 Aug 2026) in Section 1, Conclusion