= One-magnon Bloch states on a ring
{title2=$E(q)-E_0=J[1-\cos(qa)]$}
For a periodic spin-one-half <Heisenberg model> $H=-(J/\hbar^2)\sum_{n=1}^N\mathbf S_n\cdot\mathbf S_{n+1}$, $J>0$, the fully aligned state has $E_0=-NJ/4$. In its <single-excitation subspace>, let $|n\rangle$ denote a down spin at site $n$ in an otherwise up background. Direct action of each bond gives
$$
H|n\rangle=(E_0+J)|n\rangle-\frac J2(|n-1\rangle+|n+1\rangle).
$$
The normalized translation eigenstates are $|q\rangle=N^{-1/2}\sum_ne^{iqna}|n\rangle$, with $q=2\pi\ell/(Na)$ modulo $2\pi/a$. Their <energy eigenvalues> are $E_0+J[1-\cos(qa)]$. This is an exact one-magnon calculation, not a many-magnon harmonic approximation. Near $q=0$ the energy above $E_0$ is $Ja^2q^2/2$.
The wavenumber restriction comes from requiring a periodic plane-wave coefficient pattern, equivalently requiring this vector to be an eigenvector of a one-site translation with eigenvalue $e^{-iqa}$. The identity $T^N=I$ alone imposes no restriction on a general state: it already holds throughout the ring Hilbert space. An arbitrary finite sum with coefficients $e^{iqna}$ still returns after $N$ translations, but need not be a one-site translation eigenstate unless $e^{iqNa}=1$.
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