The Holstein model has been widely accepted as a model of electrons interacting with the phonons. Analysis of its ground states was accomplished decades ago. However obtained results did not completely take account of the repulsive Coulomb interactions. Recent progress has made it possible to treat such interactions rigorously. In this paper, we study the Holstein-Hubbard mdoel with the repulsive Coulomb interactions. Ground state properties of the model are investigated. Especially, the ground state of the Hamiltonian is proven to be unique for an even number of electrons on bipartite connected lattice. In addition, an infrared bound on the two-point function is given. The effects of the repulsive Coulomb interaction induce several technical difficulties.
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.
The two-dimensional many-body Holstein-Hubbard model in the
T=0 normal state is examined within the framework of the self-consistent coupling of charge fluctuation correlations to the vibrational ones. The parameters of our model are the adiabaticity, the electron concentration, as well as the electron-phonon and the Coulomb interaction strengths. A fluctuation-based effective cumulant approach is introduced to examine the
T=0 normal-state fluctuations and an analytic approximation to the true dynamical entangled ground state is suggested. Our results for the effective charge-transfer amplitude, the ground state energy, the fluctuations in the phonon population, the phonon softening as well as the coupling constant renormalizations suggest that, the recent numerical calculations of de Mello and Ranninger (Ref. 5), Berger, Val\'a\ifmmode \check{s}\else \v{s}\fi{}ek, and von der Linden (Ref. 2), and Marsiglio (Refs. 4 and 8) on systems with finite degrees of freedom can be qualitatively extended to the systems with large degrees of freedom.