Flow equations for electron-phonon interactions: phonon damping

A recently proposed method of a continuous sequence of unitary transformations will be used to investigate the dynamics of phonons, which are coupled to an electronic system. This transformation decouples the interaction between electrons and phonons. Damping of the phonons enters through the observation, that the phonon creation and annihilation operators decay under this transformation into a superposition of electronic particle-hole excitations with a pronounced peak, where these excitations are degenerate in energy with the renormalized phonon frequency. This procedure allows the determination of the phonon correlation function and the spectral function. The width of this function is proportional to the square of the electron-phonon coupling and agrees with the conventional result for electron-phonon damping. The function itself is non-Lorentzian, but apart from these scales independent of the electron-phonon coupling.