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Acta Physica Polonica A

On the Anharmonic Polaronic Model

Jerzy Konior

Published Nov 1, 1992• 0 citations
Publisher

Abstract

Anharmonic polaronic model is considered.The model consists of the electronic subsystem, described within the extended Hubbard model, coupled to anharmonic local phonons.By the canonical Holstein-Lang-Firsov transformation the electron-phonon interaction term is ruled out and, then, the effective electronic Hamiltonian is obtained by averaging over the ground state of the phononic subsystem.It is shown that anharmonicity introduces two main additional factors as compared with the harmonic case.First, the band narrowing factor is less rapidly decaying function of the electron-phonon interaction strength.Second, anharmonicity introduces further renormaliza - tion of the on-and intersite interaction between fermions.

DOI
10.12693/aphyspola.82.773

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Anharmonic polaronic model is considered.The model consists of the electronic subsystem, described within the extended Hubbard model, coupled to anharmonic local phonons.By the canonical Holstein-Lang-Firsov transformation the electron-phonon interaction term is ruled out and, then, the effective electronic Hamiltonian is obtained by averaging over the ground state of the phononic subsystem.It is shown that anharmonicity introduces two main additional factors as compared with the harmonic case.First, the band narrowing factor is less rapidly decaying function of the electron-phonon interaction strength.Second, anharmonicity introduces further renormaliza - tion of the on-and intersite interaction between fermions.
Effects of cubic and quartic anharmonic phonons on the polaronic properties are investigated in the one-dimensional Hubbard–Holstein model at half filling, using the variable-displacement Lang–Firsov canonical transformation and the exact solution of Lieb and Wu. Although the quartic anharmonicity always reduces the effect of the Coulomb repulsion U , the cubic one mainly responsible for the thermal expansion of lattice brings about the effect of strong asymmetric nature in the sign of the electron–phonon coupling constant g ; it enhances the effect of U for positive g , while it suppresses for negative g . As a result, the overall features of the anharmonic system are basically similar to those in the harmonic one for positive g , but even in the range of the realistic magnitudes for those anharmonicities, they may become qualitatively so different for negative g as to provide a first-order phase transition.
We investigate the interaction of correlated electrons with acoustical phonons using the extended \nHubbard–Holstein model. The Lang–Firsov canonical transformation allows one to obtain \nmobile polarons for which a new diagram technique and generalized Wick’s theorem is used. The \nphysics of emission and absorption of the collective phonon-field mode by the polarons is discussed \nin detail. In the strong-coupling limit of the electron-phonon interaction, and in the normal as \nwell as in the superconducting phase, chronological thermodynamical averages of products of \nacoustical phonon-cloud operators can be expressed by the products of one-cloud operator averages. \nWhile the normal one-cloud propagator has the form of a Lorentzian, the anomalous one is of \nGaussian form and considerably smaller. We have established the Mott–Hubbard and superconducting \nphase transitions in this model.
The interaction of strongly correlated electrons with phonons in the framework of the Hubbard-Holstein model is investigated. The electron-phonon interaction is considered to be strong and is an important parameter of the model, in addition to the Coulomb repulsion of electrons and the band filling. This interaction with nondispersive optical phonons is transformed to the problem of mobile polarons using the canonical transformation of Lang and Firsov. We discuss the case where the on-site Coulomb repulsion is exactly canceled by the phonon-mediated attractive interaction. It is suggested that polarons exchanging phonon clouds can lead to polaron pairing and superconductivity. The fact that the frequency of the collective mode of phonon clouds is larger than the bare frequency then determines the superconducting transition temperature.
We investigate the interaction of strongly correlated electrons with phonons in the frame of the Hubbard-Holstein model. The electron-phonon interaction is considered to be strong and is an important parameter of the model besides the Coulomb repulsion of electrons and band filling. This interaction with the nondispersive optical phonons has been transformed to the problem of mobile polarons by using the canonical transformation of Lang and Firsov. A new diagram technique is used in order to handle the strong Coulomb repulsion of the electrons, which leads to a closed set of integral equations for the irreducible on-site many-particle Green's functions. The coupling to the phonons leads to an essential renormalization of the one- and two-particle irreducible Green's functions. It also influences the parameters which control the metal-insulator or superconducting phase transition. In addition we find a collective mode of phonon clouds which surround the polarons. The physics of the emission and absorption of this mode by the polarons has been investigated.
A new two-band polaronic Hamiltonian is derived from a generalized periodic Anderson model. This is achieved by making use of a new generalized Lang–Firsov transformation on the periodic Anderson model completed with new terms representing the interactions between local phonons and both single electrons and pairs of them with opposite spins residing on the same lattice site. The novelty consists in the appearance of three- and four-fermion terms in the resulting Hamiltonian. The interaction between phonons and pairs of electrons with opposite spins was introduced at first time by Hirsch in order to describe the weakening of the Coulomb repulsion acting between two electrons with opposite spins occupying the same atomic orbital observed in experiments. The three-fermion terms represent the local interactions between a single polarons from one band and a pair of them with opposite spins from the other one. Whether it is attractive or repulsive depends on the signs of couplings of phonons to electron pairs. The four-fermion term stands for the local attraction between pairs of electrons with opposite spins from two bands. In the case of presence of such an interaction, the formation of local fermion quartets is possible. A simplified version of the generalized Hamiltonian is investigated in the end of the paper. The influence of the three- and the four-fermion terms on the properties of the two-band fermion system with two order parameters is examined. Expressions for the order parameters at zero temperatures and finite ones along with the critical temperatures for two-electron subsystems have been derived. This was achieved in the atomic limit by making use of the mean field method.
In this paper we have studied the properties of a two-fermion system interacting with the phonon field in the framework of the Holstein Hubbard model in one, two and three dimensions. We have chosen the modified Lang-Firsov variational wave function with on-site and nearest-neighbour lattice distortion for the phonon subsystem to obtain an effective electronic Hamiltonian. This effective electronic Hamiltonian is then solved exactly to obtain the binding energy and effective mass of the bipolaron. We investigated the stability of the bipolaronic phase by comparing the ground-state energy of a bipolaron and two free polarons. We observed a critical repulsive on-site Hubbard interaction U c below which the bipolaronic phases are stable for a fixed electron-phonon coupling g . In the absence of on-site repulsion, bipolaronic phases are stable over the entire range of electron-phonon coupling for one dimension, whereas there is a critical electron-phonon coupling g c for formation of a stable bipolaron in two and three dimensions.