Ground-state properties of the two-site Hubbard-Holstein model: an exact solution

We study the two-site Hubbard-Holstein model by using an extended phonon coherent state. For the nontrivial singlet bipolarons, the double occupancy probability, the fidelity and the entanglement entropy are calculated to characterize the ground-state properties in both two-site and single-site bipolaron-dominated regimes. We use the localized minimum of the fidelity to define a crossover and plot the bipolaron phase diagram, which separates the large and small entanglement region. Furthermore, the relation between the bipolaron entanglement and the correlation functions demonstrates that the large entanglement corresponds to the large magnitude of lattice deformations induced by electrons.