Effect of electron correlation on phonons in a strongly-coupled electron-phonon system

An asymptotically exact result is obtained for the renormalized phonon energy as a function of the on-site Coulomb repulsion $U_{ee}$ in the half-filled Hubbard-Holstein model in the strong-coupling region at zero temperature. The result is obtained on the basis of the effective Hamiltonian derived in the antiadiabatic region with due attention to the transition from CDW- to SDW-regime with the increase of $U_{ee}$. Somewhat contrary to naive expectation, the phonon energy is found to shift in the insulating phase and the shift becomes larger in the SDW regime than that in the CDW regime.