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Journal of Superconductivity and Novel Magnetism

Superconductivity and Hybridization in the Fermion System at Zero Temperature

P. Tarasewicz

Published Oct 18, 2011• 1 citation
Publisher

Abstract

Some ground state properties of a fermion system are investigated. An initial Hamiltonian being a generalization of the lattice Anderson model describes this system as a two-band electron-phonon one where both of electron bands are narrow (one is wider than the other). Two bands are hybridized and electrons from both of them are strongly coupled to the same local phonons. After performing the Lang–Firsov type transformation, a new polaron Hamiltonian is obtained. As a result, there appear three effective polaron-polaron interactions in this Hamiltonian and the bandwidths of both of electron bands are reduced. It is assumed that the reduction of the bandwidth of one of the bands is sufficiently strong to make this band dispersionless. Each of the interactions can lead to superconductivity on its own but only one of them is taken into account at this stage; it is the interaction between electrons from the band with a dispersion. The Bogolyubov’s mean field approach is used in order to find the spectrum of the reduced effective Hamiltonian. The effect of the hybridization on the superconducting gap and the relationship between the critical hybridization and the average number of electrons from the wider band per lattice site are investigated.

DOI
10.1007/s10948-011-1318-y

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Some ground state properties of a fermion system are investigated. An initial Hamiltonian being a generalization of the lattice Anderson model describes this system as a two-band electron-phonon one where both of electron bands are narrow (one is wider than the other). Two bands are hybridized and electrons from both of them are strongly coupled to the same local phonons. After performing the Lang–Firsov type transformation, a new polaron Hamiltonian is obtained. As a result, there appear three effective polaron-polaron interactions in this Hamiltonian and the bandwidths of both of electron bands are reduced. It is assumed that the reduction of the bandwidth of one of the bands is sufficiently strong to make this band dispersionless. Each of the interactions can lead to superconductivity on its own but only one of them is taken into account at this stage; it is the interaction between electrons from the band with a dispersion. The Bogolyubov’s mean field approach is used in order to find the spectrum of the reduced effective Hamiltonian. The effect of the hybridization on the superconducting gap and the relationship between the critical hybridization and the average number of electrons from the wider band per lattice site are 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.
We study a generalized periodic Anderson model with on-site hybridization between wide- and narrow-band electrons and strong and local coupling with the lattice deformation. Provided that the interaction with the lattice is strong enough, the narrow-band electrons will be turned into small polarons which interact attractively with each other over short distances, leading to the formation of local pairs of narrow-band electrons. This leads to a pinning of the Fermi level which is due to the fact that narrow-band electrons exist only in pair states. By means of a generalized Schrieffer-Wolff transformation we eliminate hybridization and obtain an effective Hamiltonian which describes a contact interaction between local pairs and wide-band electrons as well as the direct hopping of local pairs and interparticle Coulomb interactions. In such a system the two types of mechanisms which can lead to superconductivity have been studied. The first one is due to direct local pair hopping and involves exclusively the narrow-band subsystem giving rise to a superconductivity analogous to superfluidity in 4He^{4}\mathrm{He}4He. The second one is due to a contact interaction between local pairs and pairs of wide-band electrons. This leads to a superconducting state involving both subsystems where the local pairs of the narrow-band subsystem induce Cooper pairing amongst the electrons of the wide-band subsystem. Consequently, the single-particle spectrum of the wide-band electrons opens up a gap around the position of the narrow band of electrons in pair states.We study the phase diagrams and the superconducting properties of two such coupled subsystems involving bosons and fermions as a function of the position of the narrow band with respect to the wide band (or the relative concentration of narrow and wide-band electrons). We also show that the critical temperature Tc{T}_{c}Tc​ and Tc{T}_{c}Tc​/Eg{E}_{g}Eg​(0) [Eg{E}_{g}Eg​(0) being the energy gap] follow, in general, a strong and nonmonotonic variation. One of the most striking features of this system is that the pairs of narrow-band electrons exist in the normal phase and condense on approaching Tc{T}_{c}Tc​ from above. The existence of such a narrow boson band together with a fermion band in the normal phase leads to characteristic behavior in the specific heat over temperature variation being very similar to that observed in recent studies of nonclassical superconductors.
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.
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.
The two-band model describing strongly hybridized degenerated electron states in a chain is proposed and solved exactly by means of the Bethe ansatz. The fermions of the subbands interact with each other via one-particle and one-particle correlated on-site hybridized interactions. The chain of electrons is reduced to the model with two noninteracting subbands with a given total number of electrons, in which fermions interact via different effective constants of interactions. In the case of strong hybridized interaction the on-site repulsive interaction is compensated by hybridized interaction, which leads to an effective attractive interaction between particles and the formation of spinless bound states of Cooper type. This electron liquid component determines both heavy fermion- and superconducting-type behavior of electron properties.
A modified Anderson lattice is proposed, whose the Hamiltonian accounts for the two-particle hybridization of localized and itinerant electrons instead of one-particle hybridization which takes into account in the original Anderson model. The 1D version of this model can be solved using the Bethe ansazt. It has been shown that hybridization between itinerant and frozen localized electrons results in an effective on-site interaction between itinerant electrons. The magnitude and sign of this interaction depend on the position of the level relative to the Fermi energy (the energy of this level corresponds to the two-particle state of localized electrons at a site). It is shown that when the energy of an localized electron in the two-particle state lies above the Fermi energy, the effective on-site interaction between itinerant electrons is attractive. A repulsive interaction between itinerant electrons occurs when this energy lies below the Fermi energy. Thus, two-part hybridization between localized and itinerant electrons can lead to effective attraction between itinerant electrons, which is unique in itself and may underlie the nature of high-temperature superconductivity.