Local conserved quantities in the spin-1 XXZ chain without biquadratic exchange

Analyze the local conserved quantities of the U(1)-symmetric spin-1 XXZ chain with a single-ion anisotropy and no biquadratic exchange, corresponding to the coupling pattern f_0=f_{\pm1}=f_{\pm2}=0, and determine whether nontrivial local conserved quantities exist.

Background

The main theorem covers six patterns in which at least one but not all of the quadrupolar couplings vanishes, while the previously studied analysis covers the case in which all quadrupolar couplings are nonzero. The only coupling pattern not covered by either analysis is the spin-1 XXZ chain with single-ion anisotropy and no biquadratic exchange. The paper states that this case requires a separate analysis and remains unresolved.

References

The only pattern not covered by this work or by Ref. is the one with $f_0=f_{\pm1}=f_{\pm2}=0$, i.e., the spin-$1$ XXZ chain with a single-ion anisotropy but without biquadratic exchange, which requires a separate analysis and is left 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 2, Model, immediately after the list of six vanishing-coupling patterns; reiterated in Section 1, Conclusion