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Emergence of states in the phonon spectral function of the Holstein polaron below and above the one-phonon continuum

We investigate the low-energy properties of the Holstein polaron through calculation of the q-dependent phonon spectral function using an improved exact-diagonalization technique, defined over a variational Hilbert space. We perform a comprehensive study of the low-energy excitations of the polaron. Beside the energy range, where the additional phonon excitation is unbound, we observe separate coherent peaks which correspond to bound and antibound states of a polaron and additional phonon quanta. These novel states can be observed for intermediate and strong electron-phonon coupling strengths, as well as below and above the unbound one-phonon excitation spectrum. A detailed investigation of their properties is presented. We find good agreement between the phonon spectral function obtained from the first-order strong-coupling perturbation theory and numerical results.