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Physical review. B, Condensed matter

Spectral weight function in the Hubbard model for a cubic cluster

L. Tan, Qiming Li, J. Callaway

Published Jul 1, 1991• 15 citations
Publisher

Abstract

We report the results of a calculation of the spectral weight function of a single hole (or electron) placed in an otherwise half-filled band described by the one-orbital Hubbard model. The calculation is made by exact diagonalization. An eight-site cubic cluster is considered. The density of states is studied as the interaction strength is varied between weak and strong couplings. 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. Only modest band narrowing is found.

DOI
10.1103/physrevb.44.341

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Physical review. B, Condensed matter•Dec 15, 1994•158 citations
We report the results of a calculation of the spectral weight function of a single hole (or electron) placed in an otherwise half-filled band described by the one-orbital Hubbard model. The calculation is made by exact diagonalization. An eight-site cubic cluster is considered. The density of states is studied as the interaction strength is varied between weak and strong couplings. 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. Only modest band narrowing is found.
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 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 spectral weight function A(p,\ensuremath{\omega}) and density of states D(\ensuremath{\omega}) at half filling are calculated with an exact diagonalization technique on the 4\ifmmode\times\else\texttimes\fi{}4 square lattice in the one-band Hubbard model from weak to strong coupling. Quasiparticle peaks are observed at weak and moderate couplings, while mostly incoherent structure appears at strong coupling. The gap in the density of states can be observed for all couplings and widens as the interaction strength increases.
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.
We calculate the single-particle excitation spectrum of holes in the Emery model thereby extending and improving previous calculations. The system is considered at half filling (nh{\mathit{n}}_{\mathit{h}}nh​=1, one hole per CuO2{\mathrm{CuO}}_{2}CuO2​ unit) and for hole doping, where the on-site hole-hole repulsions are kept finite. A paramagnetic form of the ground state is used. For the determination of the retarded Green's functions of copper and oxygen holes, the projection technique is applied solving the resulting equations of motions self-consistently. At half filling, the excitation spectrum exhibits a charge-transfer gap bounded by Zhang-Rice singlet states and the upper Hubbard band. Upon hole doping the flat singlet band crosses the Fermi level giving rise to a large Fermi surface at a hole concentration of nh{\mathit{n}}_{\mathit{h}}nh​=1.25. Moreover, spectral weight is shifted from the upper Hubbard band to the states near the Fermi energy. The calculated spectral densities, the singlet dispersion for the doped system, and the transfer of spectral weight are in good quantitative agreement with exact diagonalization results for 2\ifmmode\times\else\texttimes\fi{}2 CuO2{\mathrm{CuO}}_{2}CuO2​ cluster.
Using strong-coupling perturbation theory for the Hubbard model, explicit expressions are obtained for the integrated weights, energy positions, and widths of the upper and lower Hubbard bands separately, both for the optical and photoemission spectrum. In one dimension all expressions can be explicitly evaluated using the large-U Bethe-ansatz wave function. The k\ensuremath{\rightarrow}-dependent moments for the one-particle spectrum are compared with the closely related two-pole ansatz. The strong momentum dependence of the spectra demonstrates the importance of the k\ensuremath{\rightarrow}-dependent correlation functions in the spectral moments, which are often neglected. In order to estimate corrections due to the neglect of higher-order terms (in t/U), all results are compared with numerical-diagonalization data for intermediate-U values. The sum-rule results show a rapid spectral-weight transfer from the high- to the low-energy regime upon doping, similar to what is observed experimentally in one-particle and optical spectra of Cu (high-Tc{\mathit{T}}_{\mathit{c}}Tc​) and Ni compounds. Such fast weight redistributions as a function of the carrier density are a natural consequence of the strong correlations in these materials. The redistribution of intensity in the optical conductivity is connected with the dilution of the spin system by the added holes. For the one-particle spectrum the decrease of weight in the upper Hubbard band away from half filling can be understood by state counting at U=\ensuremath{\infty}, and the effect is strongly enhanced for small doping by both the first- and second-order terms in t/U. Altogether, the one-particle and optical spectra show the importance of the three-site hopping term. Therefore the t-J model does not represent well the spectral properties of the Hubbard model at large U. The local spin order is essential and determines both the k\ensuremath{\rightarrow} dependence of the one-particle spectrum and the intensities in the optical spectrum.