  While the concept of the entanglement spectrum has heretofore been utilized
to address various many-body systems, the models describing an itinerant
spinless-fermion excitation coupled to zero-dimensional bosons (e.g.
dispersionless phonons) have as yet not received much attention in this regard.
To fill this gap, the ground-state entanglement spectrum of a model that
includes two of the most common types of short-ranged, nonlocal
excitation-phonon interaction -- the Peierls- and breathing-mode couplings --
is numerically evaluated here. This model displays a sharp, level-crossing
transition at a critical coupling strength, which signifies the change from a
nondegenerate ground state at the quasimomentum $K_{\textrm{gs}}=0$ to a
twofold-degenerate one corresponding to a symmetric pair of nonzero
quasimomenta. Another peculiarity of this model is that in the special case of
equal Peierls- and breathing-mode coupling strengths the bare-excitation Bloch
state with the quasimomentum $0$ or $\pi$ is its exact eigenstate. Moreover,
below a critical coupling strength this state is the ground state of the model.
Thus, the sharp transition between a bare excitation and a heavily
phonon-dressed (polaronic) one can be thought of as a transition between
vanishing and finite entanglement. It is demonstrated here that the smallest
ground-state entanglement-spectrum eigenvalue to a large extent mimics the
behavior of the entanglement entropy itself and vanishes in this special case
of the model; by contrast, all the remaining eigenvalues diverge in this case.
The implications of excitation-phonon entanglement for $W$-state engineering in
superconducting and neutral-atom-based qubit arrays serving as analog
simulators of this model are also discussed.
