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arXiv:1612.07188 · Condensed Matter Physics

Properties and temperature evolution of the spectrum of localized quasi-particles interacting with polarization phonons in two models

N. V. Tkach, Julia O. Seti, Voitsekhivska, Olesya Pytiuk

Published Dec 1, 2016• 7 citations
arXivHTMLPDF

Abstract

Using the Feynman-Pines diagram technique, the energy spectrum of localized quasi-particles interacting with polarization phonons is calculated and analyzed in the wide range of energies at the finite temperature of the system.It is established that the general model of the system, besides the bound states known from the simplified model with an additional condition for the operator of quasi-particles number, contains the new bound states even for the systems with weak coupling.The contribution of multi-phonon processes into the formation of renormalized spectrum of the system is analyzed.The reasons of the appearance, behaviour and disappearance of separate pairs of bound states depending on the coupling constant and temperature are revealed.

DOI
10.5488/cmp.19.43701

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Using the Feynman-Pines diagram technique, the energy spectrum of localized quasi-particles interacting with polarization phonons is calculated and analyzed in the wide range of energies at the finite temperature of the system.It is established that the general model of the system, besides the bound states known from the simplified model with an additional condition for the operator of quasi-particles number, contains the new bound states even for the systems with weak coupling.The contribution of multi-phonon processes into the formation of renormalized spectrum of the system is analyzed.The reasons of the appearance, behaviour and disappearance of separate pairs of bound states depending on the coupling constant and temperature are revealed.
The theory of renormalized energy spectrum of localized quasi-particle interacting with polarization phonons at finite temperature is developed within the Feynman-Pines diagram technique.The created computer program effectively takes into account multi-phonon processes, exactly defining all diagrams of mass operator together with their analytical expressions in arbitrary order over the coupling constant.Now it is possible to separate the pole and non-pole mass operator terms and perform a partial summing of their main terms.The renormalized spectrum of the system is obtained within the solution of dispersion equation in the vicinity of the main state where the high-and low-energy complexes of bound states are observed.The properties of the spectrum are analyzed depending on the coupling constant and the temperature.
The theory of renormalized energy spectrum of a multi-level quasiparticle interacting with polarization phonons at T=0T=0T=0 K is developed within the Feynman-Pines diagram technique in a new approach. It permits a successive separation of multiplicative diagrams from non-multiplicative ones for all orders of mass operator, and their partial summing as well. The obtained mass operator is presented as a sum of branched chain fractions, which effectively take into account the multi-phonon processes. For the two-level quasiparticle it is shown that just the interlevel (non-diagonal) interaction with phonons fundamentally changes the properties of the spectrum. In the vicinity of all threshold energies, the quasi-equidistant phonon satellite bands (i.e., groups of energy levels) are formed. They correspond to the complexes of bound states of a quasi-particle with many phonons.
Partial summing of infinite range of diagrams for the two-phonon mass operator of polaron described by Fröhlich Hamiltonian is performed using the Feynman-Pines diagram technique.Renormalized spectral parameters of ground and first excited (phonon repeat) polaron state are accurately calculated for a weak electron-phonon coupling at T = 0 K.It is shown that the stronger electron-phonon interaction shifts the energy of both states into low-energy region of the spectra.The ground state stays stationary and the excited one decays at a bigger coupling constant.
Within unitary transformed Hamiltonian of Fröhlich type, using the Green's functions method, exact renormalized energy spectrum of quasiparticle strongly interacting with two-mode polarization phonons is obtained at T=0 K in a model of the system with limited number of its initial states. Exact analytical expressions for the average number of phonons in ground state and in all satellite states of the system are presented. Their dependences on a magnitude of interaction between quasiparticle and both phonon modes are analyzed.
A finite temperature system of degenerate bosons is studied with the aid of the extended Bogoliubov transformation which has been proposed in a preceding part I. The energies of quasi-particles and the vertex functions describing their interaction are determined in an approximation which includes the lowest order effects of the interaction between non-zero momentum bosons. The quasi-particle energy depends on the distribution of thermal excitations, and is of phonon type in the long wavelength limit. The phonon nature of the spectrum is preserved by the interaction between quasi-particles, because the vertex functions vanish in the long wavelength limit. In the limit as the condensate is depleted completely, the proposed formalism simplifies to the Hartree-Fock theory.
A general iteration scheme for the many-mode degrees of freedom in locally coupled electron-phonon systems is presented. A new spectrum of phonon modes is derived and arranged into a linear chain where only the first model is vibronically coupled to the impurity, while the rest interact successively with the nearest neighbours in the chain. By dealing with a sequence of a finite number of modes along the chain an approximation scheme is developed by retaining the lower states at each step, in the spirit of quantum renormalisation-group calculations. Ground-state energies and densities of vibronic states are calculated for two-level systems for a valence-trapped mixed-valence molecular to examine the feasibility of the scheme in practice. Good convergence is achieved for both these properties. The value of the scheme for the calculation and interpretation of multi-mode effects in other electron-phonon systems is emphasised.