Excitation-number conservation in XXZ spin chains ($[H,{\sum }_n{Z}_n]=0$)

For $H={\sum }_{m<n}{c}_{mn}(X_m{X}_n+Y_m{Y}_n)+{\gamma }_{mn}{Z}_m{Z}_n$, Pauli commutators give cancellation between the $XX$ and $YY$ terms in $[H,Z_m+Z_n]$, while each $ZZ$ term commutes individually. Hence total spin magnetization is conserved. Local $Z_k$ controls preserve the same law, so dynamics split into fixed-excitation sectors even when a particular sector is controllable.