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arXiv:cond-mat/0205006 · Physical review. B, Condensed matter

Quasiparticle undressing in a dynamic Hubbard model: Exact diagonalization study

J. E. Hirsch

Published Aug 12, 2002• 25 citations
arXivHTMLPDF

Abstract

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-${T}_{c}$ 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.

DOI
10.1103/physrevb.66.064507

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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-${T}_{c}$ 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.
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.
A one-dimensional model of electrons locally coupled to spin-1/2 degrees of freedom is studied by numerical techniques. The model is one in the class of dynamic Hubbard models that describe the relaxation of an atomic orbital upon double-electron occupancy due to electron-electron interactions. We study the parameter regime where pairing occurs in this model by exact diagonalization of small clusters. World-line quantum Monte Carlo simulations support the results of exact diagonalization for larger systems and show that the kinetic energy is lowered when pairing occurs. The qualitative physics of this model and others in its class, obtained through approximate analytic calculations, is that superconductivity occurs through hole undressing even in parameter regimes where the effective on-site interaction is strongly repulsive. Our numerical results confirm the expected qualitative behavior and show that pairing will occur in a substantially larger parameter regime than predicted by the approximate low-energy effective Hamiltonian.
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 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.
The Hubbard model with occupation-dependent hopping rate exhibits superconductivity in a wide range of parameters within mean-field (BCS) theory. Here we study pair binding energies in this model for small clusters by exact diagonalization of the Hamiltonian. The model is defined by on-site and nearest-neighbor repulsions U and V, and occupation-dependent hopping rate t(n)=t+n\ensuremath{\Delta}t. We present results for one-dimensional chains and for two-dimensional square lattices of sizes 4\ifmmode\times\else\texttimes\fi{}4, 6\ifmmode\times\else\texttimes\fi{}6, and 8\ifmmode\times\else\texttimes\fi{}8. As a function of carrier density n the pair binding energy first increases and then decreases, and vanishes beyond a critical density. BCS results for the pair binding energy are found to be in remarkably good agreement with the exact results. In particular, BCS theory accurately reproduces the range of interaction parameters where pair binding exists in the exact solution. The model is found to exhibit no tendency to phase separation or clustering of more than two particles even for parameters giving rise to strong pair binding, in contrast to other tight-binding models where pairing occurs.
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.