DYNAMICAL MEAN-FIELD THEORY OF THE SMALL POLARON

A dynamical mean-field theory of the small polaron problem is presente d, which becomes exact in the limit of infinite dimensions. The ground -state properties and the one-electron spectral function are obtained for a single electron interacting with Einstein phonons by a mapping o f the lattice problem onto a polaronic impurity model. The one-electro n propagator of the impurity model is calculated through a continued f raction expansion, at both zero and finite temperature, for any electr on-phonon coupling and phonon energy. In contrast to the ground-state properties, such as the effective polaron mass, which show a continuou s behavior as the coupling is increased, spectral properties exhibit a sharp qualitative change at low enough phonon frequency: beyond a cri tical coupling, one energy gap and then more open in the density of st ates at low energy, while the high-energy part of the spectrum is broa d and can be qualitatively explained by a strong coupling adiabatic ap proximation. As a consequence, narrow and coherent low-energy subbands coexist with an incoherent featureless structure at high energy. The subbands denote the formation of quasiparticle polaron slates. Also, d ivergencies of the self-energy may occur in the gaps. At finite temper ature such an effect triggers an important damping and broadening of t he polaron subbands. On the other hand, in the large phonon frequency regime such a separation of energy scales does not exist and the spect rum always has a multipeaked structure.