Integrability from unconventional boosts without the [X,[B,X]]=0 condition

Determine whether, for translationally invariant quantum spin chains with finite-range interactions, the existence of a Hermitian at-most-two-local boost operator B that generates a nonzero conserved quantity Q^(3) = [B, H] suffices to imply integrability even when the translation-defect operator X := T(B) − B does not satisfy the commutation condition [X, [B, X]] = 0. Specifically, ascertain whether the iteratively defined family of charges Q^(n+1) := [B, Q^(n)] yields infinitely many mutually commuting local conserved quantities under this violation of the commutator constraint.

Background

The paper proves that the Reshetikhin condition—equivalent to the conservation of the third-order charge generated by the canonical boost—implies full integrability by producing an infinite hierarchy of commuting local charges. It further establishes the Grabowski–Mathieu conjecture for general boosts in models satisfying specific structural assumptions, and provides a broader criterion (Proposition and Theorem statements) ensuring integrability when a certain commutator constraint is met.

Beyond these proven regimes, the authors identify a gap: a class of “unconventional” boost operators may generate conserved quantities but fail the commutation condition [X, [B, X]] = 0, where X is the translation defect X = T(B) − B. Whether such boosts nonetheless guarantee integrability remains explicitly unknown and is highlighted as an open direction.