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Physical review. B, Condensed matter

Spectral function in the two-dimensional Hubbard model

Qiming Li, J. Callaway, L. Tan

Published Nov 1, 1991• 13 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.44.10256

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The spectral functions of one hole in the t-J and one-band Hubbard models are calculated using exact diagonalization techniques on small lattices. Results for the t-${\mathit{J}}_{\mathit{z}}$ model are also presented. For the t-J model we found that there is a quasiparticle at the bottom of the hole spectrum with an energy well approximated by ${\mathit{E}}_{\mathit{h}}$=-3.17+2.83${\mathit{J}}^{0.73}$ (for 0.1\ensuremath{\le}J\ensuremath{\le}1.0, t=1) on a 4\ifmmode\times\else\texttimes\fi{}4 lattice. The rest of the spectrum is not incoherent: We identified at least two other peaks following a similar power-law behavior with J. We speculate that the J dependence of the results can be explained by a model where the hole is trapped in a confining potential as in the Ising limit. The bandwidth of the hole is linear in J in the region 0.1\ensuremath{\le}J\ensuremath{\le}0.4 although a power-law behavior is not excluded. The spectral weight of the quasiparticle grows like ${\mathit{J}}^{0.5}$ in the same region. We present new analytical results in the large J/t limit to understand the motion of the hole: In perturbation theory it can be shown that the momentum of the hole at large J/t is k=(\ensuremath{\pi},\ensuremath{\pi}) changing to k=(\ensuremath{\pi}/2,\ensuremath{\pi}/2) at intermediate J/t in agreement with numerical and spin-waves results. We show analytically and numerically that the bandwidth of the quasiparticle is of order t in the large J/t limit. This result corresponds to a spin-liquid state. The one-hole spectral function of the Hubbard model is obtained for lattices with 8 and 10 sites. A quasiparticle is also observed in this case. The bandwidth and the relation with the t-J model are discussed and a comparison with recent Monte Carlo results is made. We also review and extend previous results for the ground-state properties of the t-J model.

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arXiv:cond-mat/0207369•Physical review. B, Condensed matter•Jan 6, 2003•32 citations
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.
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 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.
The self-energy of the hole-doped two-dimensional (2D) Hubbard model is calculated to second order in the interaction U and the ensuing renormalization of the spectral properties and of the Fermi surface is discussed. In uncorrelated systems the square-shaped Fermi surface separates the Fermi surface closed around the \ensuremath{\Gamma} point (with more holes than particles) from the Fermi surface closed around the Z point (with more particles than holes). In a correlated system a topological change from a Fermi surface centered around the \ensuremath{\Gamma} point to a Fermi surface closed around a Z point is induced, either by increasing the interaction or by diminishing the concentration of holes. The shape of the renormalized spectral function Ap{\mathit{A}}_{\mathbf{p}}Ap​(\ensuremath{\omega}) is momentum dependent. Using Ap{\mathit{A}}_{\mathbf{p}}Ap​(\ensuremath{\omega}) we evaluate dispersion of single-particle excitations. At low energies the band of quasiparticles with reduced bandwidth is clearly seen. At high energies and far away from the Fermi surface the spectra acquire an additional peak that describes excitations in the Hubbard band. The dispersion in Hubbard bands is weaker along the \ensuremath{\Gamma}-Z, than \ensuremath{\Gamma}-X or X-Z direction. The density of states of a correlated system, as given by the perturbation theory, remains logarithmically singular but the singular weight is reduced with respect to the uncorrelated one. In addition, the correlations transfer the spectral weight out of the low-energy region, where only a narrow Kondo-like structure seems to remain. The spectral properties of the 2D Hubbard model obtained by the truncated perturbation expansion resemble in many ways the recent experimental data on metallic cuprates.
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 functions of one hole in the t-J and one-band Hubbard models are calculated using exact diagonalization techniques on small lattices. Results for the t-Jz{\mathit{J}}_{\mathit{z}}Jz​ model are also presented. For the t-J model we found that there is a quasiparticle at the bottom of the hole spectrum with an energy well approximated by Eh{\mathit{E}}_{\mathit{h}}Eh​=-3.17+2.83J0.73{\mathit{J}}^{0.73}J0.73 (for 0.1\ensuremath{\le}J\ensuremath{\le}1.0, t=1) on a 4\ifmmode\times\else\texttimes\fi{}4 lattice. The rest of the spectrum is not incoherent: We identified at least two other peaks following a similar power-law behavior with J. We speculate that the J dependence of the results can be explained by a model where the hole is trapped in a confining potential as in the Ising limit. The bandwidth of the hole is linear in J in the region 0.1\ensuremath{\le}J\ensuremath{\le}0.4 although a power-law behavior is not excluded. The spectral weight of the quasiparticle grows like J0.5{\mathit{J}}^{0.5}J0.5 in the same region. We present new analytical results in the large J/t limit to understand the motion of the hole: In perturbation theory it can be shown that the momentum of the hole at large J/t is k=(\ensuremath{\pi},\ensuremath{\pi}) changing to k=(\ensuremath{\pi}/2,\ensuremath{\pi}/2) at intermediate J/t in agreement with numerical and spin-waves results. We show analytically and numerically that the bandwidth of the quasiparticle is of order t in the large J/t limit. This result corresponds to a spin-liquid state. The one-hole spectral function of the Hubbard model is obtained for lattices with 8 and 10 sites. A quasiparticle is also observed in this case. The bandwidth and the relation with the t-J model are discussed and a comparison with recent Monte Carlo results is made. We also review and extend previous results for the ground-state properties of the t-J model.
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.