Magnetization sectors of a spin chain
= Magnetization sectors of a spin chain
{title2=$S=N-2k,\quad\dim\mathcal H_k=\binom Nk$}
With $S=\sum_{n=1}^NZ_n$, a <computational basis> vector containing $k$ excitations has <eigenvalue> $N-2k$. There are $N+1$ distinct eigenspaces, of dimensions $\binom Nk$. Any <Hamiltonian> commuting with $S$ preserves each sector and their population weights. In particular, a state supported in one sector cannot be transferred to a different sector by such controls.