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

TL;DR

This paper derives an exact expression for how electron correlation affects phonon energies in a strongly-coupled electron-phonon system, revealing unexpected phonon energy shifts in different insulating regimes.

Contribution

It provides an asymptotically exact result for phonon energy renormalization in the Hubbard-Holstein model considering electron correlation effects.

Findings

Phonon energy shifts are larger in the SDW regime than in the CDW regime.

The phonon energy shifts occur even in the insulating phase.

The results are derived using an effective Hamiltonian in the antiadiabatic limit.

Abstract

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.