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International Journal of Modern Physics B

The generalized Lang–Firsov transformation and the three- and the four-fermion interactions in a lattice fermion model

P. Tarasewicz

Published Jun 7, 2016• 4 citations
Publisher

Abstract

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.

DOI
10.1142/s0217979216501241

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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.
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.
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.
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.
Starting from the shell structure in atoms and the significant correlation within electron pairs, we distinguish the exchange-correlation effects between two electrons of opposite spins occupying the same orbital from the average correlation among many electrons in a crystal. In the periodic potential of the crystal with lattice constant larger than the effective Bohr radius of the valence electrons, these correlated electron pairs can form a metastable energy band above the corresponding single-electron band separated by an energy gap. In order to determine if these metastable electron pairs can be stabilized, we calculate the many-electron exchange-correlation renormalization and the polaron correction to the two-band system with single electrons and electron pairs. We find that the electron-phonon interaction is essential to counterbalance the Coulomb repulsion and to stabilize the electron pairs. The interplay of the electron-electron and electron-phonon interactions, manifested in the exchange-correlation energies, polaron effects, and screening, is responsible for the formation of electron pairs (bipolarons) that are located on the Fermi surface of the single-electron band.