Dynamical properties of the two-dimensional Holstein-Hubbard model in   the normal state at zero temperature: A fluctuation-based effective cumulant   approach
  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, Valasek and von der Linden (Ref.2) and Marsiglio (Ref. 4 and 8) on systems with finite degrees of freedom can be qualitatively extended to the systems with large degrees of freedom. 
T. Hakioglu and M.Ye. Zhuravlev