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arXiv:cond-mat/9912393

Evolution of Fermi Liquid Behavior with Doping in the Hubbard Model

Jungsoo Kim, Dermot Coffey

Published Dec 21, 1999• 0 citations
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Abstract

We calculate the single-particle Green's function for the tight-binding band structure, $ξ_{\vec p}=-2t\cos p_x-2t\cos p_y -μ$, with a function of chemical potential $μ$ for square-lattice system. The form of the single-particle self-energy, $Σ({\vec p}, E)$, is determined by the density-density correlation function, $χ({\vec q}, ω)$, which develops two peaks for $μ\gtrsim -2.5t$ unlike parabolic band case. Near half filling $χ({\vec q}, ω)$ becomes independent of $ω$, one dimensional behavior, at intermediate values of $ω$ which leads to one dimensional behavior in $Σ({\vec p},E)$. However $μ\leq -0.1t$ there is no influence on the Fermi Liquid dependences from SDW instability. The strong $\vec p$ and $E$ dependence of the off-shell self-energy, $Σ(p,E)$, found earlier for the parabolic band is recovered for $μ\lesssim -t$ but deviations from this develop for $μ\gtrsim -0.1t$. The resonance peak width of the spectral function, $A({\vec p}, E)$ has linear dependence in $ξ_{\vec p}$ due to the $E$ dependence of the imaginary part of $Σ({\vec p}, E)$. We point out that an accurate detailed form for $Σ({\vec p},E)$ would be very difficult to recover from ARPES data for the spectral density.

DOI
10.48550/arxiv.cond-mat/9912393

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We calculate the single-particle Green's function for the tight-binding band structure, ξp⃗=−2tcos⁡px−2tcos⁡py−μξ_{\vec p}=-2t\cos p_x-2t\cos p_y -μξp​​=−2tcospx​−2tcospy​−μ, with a function of chemical potential μμμ for square-lattice system. The form of the single-particle self-energy, Σ(p⃗,E)Σ({\vec p}, E)Σ(p​,E), is determined by the density-density correlation function, χ(q⃗,ω)χ({\vec q}, ω)χ(q​,ω), which develops two peaks for μ≳−2.5tμ\gtrsim -2.5tμ≳−2.5t unlike parabolic band case. Near half filling χ(q⃗,ω)χ({\vec q}, ω)χ(q​,ω) becomes independent of ωωω, one dimensional behavior, at intermediate values of ωωω which leads to one dimensional behavior in Σ(p⃗,E)Σ({\vec p},E)Σ(p​,E). However μ≤−0.1tμ\leq -0.1tμ≤−0.1t there is no influence on the Fermi Liquid dependences from SDW instability. The strong p⃗\vec pp​ and EEE dependence of the off-shell self-energy, Σ(p,E)Σ(p,E)Σ(p,E), found earlier for the parabolic band is recovered for μ≲−tμ\lesssim -tμ≲−t but deviations from this develop for μ≳−0.1tμ\gtrsim -0.1tμ≳−0.1t. The resonance peak width of the spectral function, A(p⃗,E)A({\vec p}, E)A(p​,E) has linear dependence in ξp⃗ξ_{\vec p}ξp​​ due to the EEE dependence of the imaginary part of Σ(p⃗,E)Σ({\vec p}, E)Σ(p​,E). We point out that an accurate detailed form for Σ(p⃗,E)Σ({\vec p},E)Σ(p​,E) would be very difficult to recover from ARPES data for the spectral density.
We calculate the single-particle Green's function for a contact interaction with nearest-neighbor hopping on a square lattice as a function of chemical potential $\ensuremath{\mu}.$ This allows us to investigate the dependence of the leading Fermi-liquid dependencies on the band structure as the Fermi surface evolves from a circle at $\ensuremath{\mu}\ensuremath{\sim}\ensuremath{-}4t$ to a square at $\ensuremath{\mu}=0.$ The form of the single-particle self-energy $\ensuremath{\Sigma}(\stackrel{\ensuremath{\rightarrow}}{p},E)$ is determined by the density-density correlation function $\ensuremath{\chi}(\stackrel{\ensuremath{\rightarrow}}{q},\ensuremath{\omega})$ which develops two peaks for $\ensuremath{\mu}\ensuremath{\gtrsim}\ensuremath{-}2.5t$ unlike the parabolic band case. Near half-filling, $\ensuremath{\chi}(\stackrel{\ensuremath{\rightarrow}}{q},\ensuremath{\omega})$ becomes independent of $\ensuremath{\omega},$ one-dimensional behavior, at intermediate values of $\ensuremath{\omega},$ which leads to the one-dimensional behavior in $\ensuremath{\Sigma}(\stackrel{\ensuremath{\rightarrow}}{p},E).$ However, with $\ensuremath{\mu}<~\ensuremath{-}0.1t$ there is no influence on the Fermi-liquid dependencies from the spin-density wave instability. We find that throughout the doping region $\ensuremath{\Sigma}(\stackrel{\ensuremath{\rightarrow}}{p},E)$ remains qualitatively the same as for the isotropic Fermi surface with quantitative differences. The strong $\stackrel{\ensuremath{\rightarrow}}{p}$ and E dependence of the off-shell self-energy $\ensuremath{\Sigma}(p,E)$ found earlier for the parabolic band is recovered for $\ensuremath{\mu}\ensuremath{\lesssim}\ensuremath{-}t$ but deviates from this develop for $\ensuremath{\mu}\ensuremath{\gtrsim}\ensuremath{-}0.1t.$ The resonance peak width of the spectral function $A(\stackrel{\ensuremath{\rightarrow}}{p}E),$ has a linear dependence in ${\ensuremath{\xi}}_{\stackrel{\ensuremath{\rightarrow}}{p}}$ due to the E dependence of the imaginary part of $\ensuremath{\Sigma}(\stackrel{\ensuremath{\rightarrow}}{p},E).$ We point out that an accurate detailed form for $\ensuremath{\Sigma}(\stackrel{\ensuremath{\rightarrow}}{p},E)$ would be very difficult to recover from angle-resolved photoemission spectroscopy data for the spectral density. Since the leading corrections are determined by the long wavelength particle-hole excitations, the results found here for the Hubbard model carry over to Hamiltonians with finite range interactions.
The electron self-energy Sigma(k,\ensuremath{\omega}) corresponding to the one-band Hubbard Hamiltonian on clusters of the square lattice has been obtained as a function of the band filling up to second order in the interaction parameter U. The k dependence of the self-energy is completely taken into account. As a consequence, the imaginary part of the self-energy shows a linear behavior for fillings close to one electron per site whereas for large doping rates the quadratic \ensuremath{\omega} dependence characterizing Fermi liquids is recovered. The origin of this behavior has been investigated analytically: linear terms associated with nesting are shown to exist in any finite-dimensional lattice although its numerical relevance decreases very rapidly for space dimensionality larger than two. Implications of these results on the renormalization factor Z have been analyzed. From a practical point of view, we conclude that a standard second-order perturbative treatment is quantitatively precise for U values of the on-site Coulomb interaction smaller than the bandwidth of the noninteracting spectrum.
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.
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We present the doping dependence of the spectral functions, density of states and low frequency behavior of the self-energy for the 2D Hubbard model on the basis of our recently developed theory for the Hubbard model. Strong 2D critical spin fluctuations dominating near half-filling are completely destroyed by 13 and 20 percent doping concentrations for U=4 and 8, respectively. Below these concentrations the imaginary part of the self-energy vanishes quadratically in frequency near the Fermi energy, a characteristic feature for the Fermi liquid.