Scalene Yang-Baxter triples realize all nearest-neighbour Hamiltonians with global additive symmetries

Abstract: We establish that bond Hamiltonian densities, up to local difference terms whose sum vanishes under periodic boundary conditions, lying in the commutant of bond operators, $w_j\pm w_{j+1}$, is a necessary and sufficient condition for a one-dimensional nearest-neighbour Hamiltonian to conserve a uniform or staggered sum of the one-site operators $w_j$. Starting from these local commutants, we show that every such Hamiltonian and its conserved charge can be obtained from monodromy matrices constructed using regular and non-regular members of scalene Yang-Baxter triples. The results hold on periodic chains of length $L\geq3$, with even length required in the staggered case. The results do not require translation invariance, and hence the interaction on each bond can be chosen independently from the full corresponding commutant. Our representation-independent construction applies in arbitrary finite local dimension and exhausts the Hamiltonians with either prescribed additive symmetry, without requiring ordinary Yang-Baxter integrability. We illustrate the construction with spin-$\tfrac12$ and spin-1 systems and discuss its relation to the double-commutant theorem. Classifying the one-site generators $w$ and their associated bond commutants provides a route to classifying such Hamiltonians with additive conserved charges.