Continuous symmetry breaking without a commuting order parameter

Characterize continuous symmetry breaking in frustration-free quantum systems that lack a local order parameter commuting with the Hamiltonian, including the mechanism by which such systems evade the conventional one-dimensional no-go arguments.

Background

The paper establishes that the frustration-free spin-1 chain studied in the main text and the periodic Motzkin chain have no nontrivial local conserved quantities in the proven locality range, and therefore no corresponding local commuting order parameter in that range. It argues that anomalously soft excitations associated with frustration-freeness may enable symmetry breaking without the conserved-order-parameter mechanism of the Heisenberg ferromagnet. A general theory of this phenomenon is explicitly left for future work.

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