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arXiv:cond-mat/0207369 · Physical review. B, Condensed matter

Electronic dynamic Hubbard model: Exact diagonalization study

J. E. Hirsch

Published Jan 6, 2003• 32 citations
arXivHTMLPDF

Abstract

A model to describe electronic correlations in energy bands is considered. The model is a generalization of the conventional Hubbard model that allows for the fact that the wave function for two electrons occupying the same Wannier orbital is different from the product of single-electron wave functions. We diagonalize the Hamiltonian exactly on a four-site cluster and study its properties as a function of band filling. The quasiparticle weight is found to decrease and the quasiparticle effective mass to increase as the electronic band filling increases, and spectral weight in one- and two-particle spectral functions is transferred from low to high frequencies as the band filling increases. Quasiparticles at the Fermi energy are found to be more ``dressed'' when the Fermi level is in the upper half of the band (hole carriers) than when it is in the lower half of the band (electron carriers). The effective interaction between carriers is found to be strongly dependent on band filling becoming less repulsive as the band filling increases and attractive near the top of the band in certain parameter ranges. The effective interaction is most attractive when the single-hole carriers are most heavily dressed, and in the parameter regime where the effective interaction is attractive, hole carriers are found to ``undress,'' hence become more like electrons, when they pair. It is proposed that these are generic properties of electronic energy bands in solids that reflect a fundamental electron-hole asymmetry of condensed matter. The relation of these results to the understanding of superconductivity in solids is discussed.

DOI
10.1103/physrevb.67.035103

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A model to describe electronic correlations in energy bands is considered. The model is a generalization of the conventional Hubbard model that allows for the fact that the wave function for two electrons occupying the same Wannier orbital is different from the product of single-electron wave functions. We diagonalize the Hamiltonian exactly on a four-site cluster and study its properties as a function of band filling. The quasiparticle weight is found to decrease and the quasiparticle effective mass to increase as the electronic band filling increases, and spectral weight in one- and two-particle spectral functions is transferred from low to high frequencies as the band filling increases. Quasiparticles at the Fermi energy are found to be more dressed'' when the Fermi level is in the upper half of the band (hole carriers) than when it is in the lower half of the band (electron carriers). The effective interaction between carriers is found to be strongly dependent on band filling becoming less repulsive as the band filling increases and attractive near the top of the band in certain parameter ranges. The effective interaction is most attractive when the single-hole carriers are most heavily dressed, and in the parameter regime where the effective interaction is attractive, hole carriers are found to undress,'' hence become more like electrons, when they pair. It is proposed that these are generic properties of electronic energy bands in solids that reflect a fundamental electron-hole asymmetry of condensed matter. The relation of these results to the understanding of superconductivity in solids is discussed.
Dynamic Hubbard models have been proposed as extensions of the conventional Hubbard model to describe the orbital relaxation that occurs upon double occupancy of an atomic orbital. These models give rise to pairing of holes and superconductivity in certain parameter ranges. Here we explore the changes in carrier effective mass and quasiparticle weight and in one- and two-particle spectral functions that occur in a dynamic Hubbard model upon pairing, by exact diagonalization of small systems. It is found that pairing is associated with lowering of effective mass and increase of quasiparticle weight, manifested in transfer of spectral weight from high to low frequencies in one- and two-particle spectral functions. This ``undressing'' phenomenology resembles observations in transport, photoemission, and optical experiments in high-Tc{T}_{c}Tc​ cuprates. This behavior is contrasted with that of a conventional electron-hole symmetric Holstein-like model with attractive on-site interaction, where pairing is associated with ``dressing'' instead of undressing.
The spectral weight functions and the optical conductivity of a highly correlated electronic system described by a simple Hubbard model are studied. The calculations are carried out for a small system, a cubic cluster of eight sites, by means of exact diagonalization using symmetrized basis functions. The spectral weight functions of a single hole (or electron) placed in an otherwise half-filled band are obtained. The density of states at different interaction strengths, ranging from weak to strong couplings, is studied. We observe the spreading of the spectral weight associated with some single-particle eigenstates over a large range of energies, and the appearance of satellite structure at higher excitation energies. The nature of band narrowing is also discussed. The frequency dependent (optical) conductivity is studied for the cases of a half-filled band, and for one hole and two hole dopings away from half filling. The interaction strength is varied between the weak- and strong-coupling limits. The formation of a Hubbard gap is observed. Results are related to aspects of the metal-insulator transition in bulk systems.
The conventional, single band Hubbard model captures the dynamics that arise in strongly correlated electron systems in the competition between electron transport and local Coulomb interactions. However, some of the important physics of real lattices is overlooked by its simplicity. The dynamic Hubbard model (DHM) provides additional degrees of freedom for electron states in a lattice by including a higher energy orbital and hybridization parameters for electrons to take advantage of a reduced Coulomb repulsion in the upper orbital when the lattice is at half-filling. This ``relaxation'' of the electronic state is accompanied by an asymmetry between electron and hole quasiparticles in the model. In this thesis we employ the computational tool of multiorbital dynamical mean field theory (MODMFT) to study electron-hole asymmetry in the DHM as evidenced in the behaviour of the quasiparticle weight and optical conductivity of electrons. In agreement with previous exact diagonalization work on the DHM, we find that the relaxation of electronic states is associated with quasiparticle dressing in the hole regime of the lower orbital, and a transfer of optical conductivity spectral weight from low to high frequencies. These observations are evidences of the electron-hole asymmetry in the two-orbital system. In this context, we evaluate the role of two critical variables in the DHM: the local and nonlocal hybridizations which parameterize interorbital electron transitions in the Hamiltonian. We demonstrate that the electron-hole asymmetry of the model has a more complex relationship to these hybridization parameters than previously identified. Mott physics plays an essential role in the quasiparticle dressing due to the local hybridization, while the nonlocal hybridization promotes electron-hole asymmetry by providing a separate channel for electron transport which serves to reduce the quasiparticle weight of holes in the lower band. We also briefly examine the phenomenon of orbital selective Mott transitions in the DHM within the parameter regime of electron state relaxation, and evaluate the role played by the hybridization parameters in this context.
The single-band Hubbard model (both with and without first-neighbor interactions) is studied by exact diagonalization on several one- and two-dimensional clusters, ranging from four to nine sites, including rings and arrangements of squares. Occupancies include the half-filled band (one electron per site) and configurations with one and two holes in the half-filled band. Ground-state properties are determined for a wide range of parameters. We discuss the phase diagram in the (U,V) plane and consider spin, charge, and pairing correlations. The binding of a pair of holes is investigated. Circumstances in which binding and pairing are enhanced by first-neighbor interactions are described.
We have studied the pair susceptibilities for all possible pair wave functions that fit on a two-dimensional (2D) eight-site Hubbard cluster by exact diagonalization of the Hamiltonian. Band fillings corresponding to four and six electrons were studied (two or four holes in the half-filled band) for a wide range of Hubbard interaction strengths and temperatures. Our results show that all pairing susceptibilities are suppressed by the Hubbard repulsion. We have also carried out perturbation-theory calculations which show that the leading-order U2{U}^{2}U2 contributions to the d-wave pair susceptibility suppresses d-wave pairing over a significant temperature range. These results are consistent with recent Monte Carlo results and provide further evidence suggesting that the 2D Hubbard model does not exhibit superconductivity.
The spectral function of a single hole in the two-dimensional Hubbard model is studied numerically through exact diagonalization for a nine-site cluster over a wide range of interaction strengths U. In the weak-interaction limit, the spectral function consists of sharp quasiparticle peaks corresponding to the allowed single-particle excitations of the noninteracting system. As the interaction strength increases, the spectral weight begins to spread over a large range of energies, and satellite structures appear at higher excitation energy. The band-narrowing effect and the quasiparticle weight are investigated. The results are discussed in connection with other analytical and numerical studies on the spectral function.