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I will try to frame my question using the transverse field Ising model in the low spin-coupling limit as motivation. I'll begin by discussing a case I believe I understand, that of eigenstates of continuous translations.


In quantum mechanics, we identify momentum as the generator of translations, which is expressed as $\hat{T}(a) = e^{-i a\hat{p}}$. Here, $\hat{p}$ refers to the total momentum of the system. For example, if we have two free particles, 1 and 2, with position wavefunctions $\Psi(x_1, x_2)=\frac{1}{2\pi}e^{-ix_1p_1}e^{-ix_2p_2}$, then $\hat{T}(a)\Psi(x_1, x_2)=\Psi(x_1+a,x_2+a)=e^{-(p_1+p_2)a}\Psi(x_1,x_2)$. Here it is very natural to read off the total momentum from the argument of the exponential on the right hand side as $p_1+p_2$.


We also often handle periodic lattices in quantum mechanics. For a concrete example, let's consider the 1D transverse field Ising chain, with Hamiltonian $H = -J\sum\limits_{<ij>}\sigma_i^z\sigma_j^z-h\sum\limits_i\sigma_i^x$ with N spins on a ring. The Hamiltonian enjoys invariance under discrete translations of $i \rightarrow i+1$, and so we can write eigenstates of the Hamiltonian as eigenfunctions of the discrete time translation operator.

Let us consider the limit of $J=0$. Then our ground state will look like $|0> =|\rightarrow \rightarrow \rightarrow \rightarrow ...>$. This ground state is clearly an eigenstate of the discrete translation operator.

We will have a collection of N-fold degenerate first excited states that look like $|i>=|\rightarrow\rightarrow\leftarrow_i\rightarrow...> $. We often identify the flipped spin "i" with a particle. We can write these as discrete translation eigenstates by considering the N states $|k> = \frac{1}{N}\sum\limits_j e^{-ikj}|j>$ for $k=-\pi+\frac{2\pi}{N},-\pi+2\frac{2\pi}{N}, ...,\pi.$

These $|k>$ states become important when considering small, nonzero $J$, as they diagonalize the perturbation in the degenerate subspace of first excited states. We typically identify $k$ as the momentum in analogy to the continuous case at top.

Now, the next highest energy levels at $J=0$ feature a $1/2N(N-1)$ degeneracy, with states that look like $|i,j> = |\rightarrow\rightarrow\leftarrow_i\rightarrow\leftarrow_j...>$. My question is how may I take linear combinations of these degenerate states to make eigenstates of the discrete translation operator and how may I associate a total momentum with those eigenstates? Is there a similar canonical case like the continuous one at top, $\hat{T}(a)\Psi(x_1, x_2)=\Psi(x_1+a,x_2+a)=e^{-(p_1+p_2)a}\Psi(x_1,x_2)$, where it is clear that the individual momenta add to form the total momenta?

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