  We study the off-diagonal matrix elements of observables that break the
translational symmetry of a spin-chain Hamiltonian, and as such connect energy
eigenstates from different total quasimomentum sectors. We consider
quantum-chaotic and interacting integrable points of the Hamiltonian, and focus
on average energies at the center of the spectrum. In the quantum-chaotic
model, we find that there is eigenstate thermalization; specifically, the
matrix elements are Gaussian distributed with a variance that is a smooth
function of $\omega=E_{\alpha}-E_{\beta}$ ({$E_{\alpha}$} are the
eigenenergies) and scales as $1/D$ ($D$ is the Hilbert space dimension). In the
interacting integrable model, we find that the matrix elements exhibit a skewed
log-normal-like distribution and have a variance that is also a smooth function
of $\omega$ that scales as $1/D$. We study in detail the low-frequency behavior
of the variance of the matrix elements to unveil the regimes in which it
exhibits diffusive or ballistic scaling. We show that in the quantum-chaotic
model the behavior of the variance is qualitatively similar for matrix elements
that connect eigenstates from the same versus different quasimomentum sectors.
We also show that this is not the case in the interacting integrable model for
observables whose translationally invariant counterpart does not break
integrability if added as a perturbation to the Hamiltonian.
