Effects Of Coupling To Two Phonon Modes On The Ground State Properties Of Holstein Polaron

A polaron is a quasi-particle in which electron and optical longitudinal phonon coupledwith each other. A charge placed in a polarizable medium will be screened. Dielectric theorydescribes the phenomenon by the induction of a polarization around the charge carrier. Theinduced polarization will follow the charge carrier when it is moving through the medium.The carrier together with the induced polarization is considered as one entity, which is calleda polaron.Major theoretical work has focused on solving Holstein Hamiltonians based on couplingto a single optical phonon mode. Actually Kornilovitch has investigated the multiple opticalphonon modes of arbitrary dispersion and polarization by analytical path-integral approach.The general formulas they obtained involved a large number of integrations and summations,which was prohibitively expensive to execute the direct calculation of the full action at everyMonte Carlo update. The practical impact of the formalism will depend on looking for aneffective numerical method in the future.The Lanczos algorithm based on the Exact Diagonalization technique has given reliablenumerical results of the Holstein model for an electron coupled to a single optical phononmode. As the efficient method can be systematically expanded to achieve high accuracy withmodest computational resources, we investigate the effects of coupling to two phonon modeson the properties of a Holstein polaron further.With comparison between the ground-state energy of a single mode and two modes, weconcluded that the quasi-particle wave function is predominantly electronic character forsmall k, phonon excitation of lower frequency will emerge as the quasi-particle momentumincreases, and then the wave function is predominantly phonon character for large momentum.We confirm the signification of the critical value of the wave vectork0through calculatingthe static correlation functionZ kandN phkin the momentum space.In the lattice space, we calculate the lattice deformation correlation function (i j)and phonon number function (i j)for a certain momentum. The deformation is limited to only a few lattice sites around the electron and the extra phonon excitation is bound to thepolaron atk k0, while atk k0, the range of the deformation and the extra phononexcitation extend over the entire region. As a result, the quasiparticle weight of electron andphonon sensitively depends on the magnitude of polaron momentum.