Eigenstates of a Hamiltonian diagonal in momentum basis Suppose we have a 1D Hamiltonian diagonal in $k$-space: $$ H = \sum_k \omega(k) \sigma_k^+ \sigma_k^-, \quad \sigma_k^- = \frac{1}{\sqrt{N}} \sum_{j=1}^{N} e^{-i k j} \sigma_j^-,$$ where $k = 0, 2\pi/N , \ldots, (N-1) \times 2\pi/N$. Is there a general construction for the $2^N$ eigenstates of such Hamiltonian?
So far I have worked out the eigenstates for all spins up and down, as well as the single-excitation states $\sigma_k^- |{\uparrow}\rangle^{\otimes N}$.
 A: Just rewrite the Hamiltonian as $H = \frac{1}{2} \, \sum_k\,  \omega (k) \, \left(\mathbb{1} + \sigma^z_k \right) = E_0 \mathbb{1} + \frac{1}{2} \sum_k \, \omega (k) \, \sigma^z_k$ and then the $2^N$ eigenstates of $H$ correspond to all possible choices of which $k$ modes are in the $+$ $Z$ state and which are in the $-$ $Z$ state. I guess you've worked out the cases with all modes unoccupied / occupied.
You can label the eigenstates by the "occupations" $n^{\,}_k = 0,1$, where $\sigma^{z}_k \, \left| n^{\,}_k \right\rangle = (-1)^{n^{\,}_k}\, \left| n^{\,}_k \right\rangle $, so that
$$H = \sum\limits_k \, \frac{1}{2} \, \omega(k) \left( 1 + \hat{n}^{\,}_k \right)$$
with corresponding eigenstates
$$\left|n^{\,}_1, \dots, n^{\,}_{N} \right\rangle$$
that specify all $2^N$ possible eigenstates, with energies above.
This is the same result as appeared in my original answer, where I referenced fermions so that the $n$ occupations would make more sense (since I prefer $n=0,1$ occupations over $\sigma=\pm 1$ up/down variables for writing energies of free models).
