Ground-state entanglement spectrum of a generic model with nonlocal excitation-phonon coupling

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.