Polaron-phonon interaction in a finite-size lattice: A perturbative approach

Abstract

The finite-size exciton-phonon system is revisited within the small polaron theory. Two strategies are used to treat the polaron-phonon interaction. The interaction is first expanded as a Taylor series with respect to the coupling strength. It describes polaron scattering mediated by the exchange of real and virtual phonons, the latter resulting from phonon vacuum fluctuations. However, this method does not improve energy calculations when compared with standard second-order perturbation theory. A more accurate approach is obtained by using a normally order expansion of the interaction. The vacuum fluctuations are renormalized up to infinity and a polaron Hamiltonian is defined in terms of inhomogeneous effective hopping constants. Due to the finite size of the lattice, the polaron energy spectrum exhibits discrete energy levels that are red shifted owing to the polaron-phonon interaction. By contrast, each phonon frequency is either red or blue shifted depending on the nature of the state occupied by the polaron that accompanies the phonon. But the larger is the lattice size, the smaller is the phonon frequency shift. Finally, for odd lattice sizes, the phonon frequencies remain unchanged when the polaron occupies the band center.