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arXiv:cond-mat/9409044 · Journal of the Physical Society of Japan

Quasiparticles in Two-Dimensional Hubbard Model: Splitting of Spectral Weight

Hidetoshi Fukuyama, Masao Ogata

Published Nov 15, 1994• 21 citations
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Abstract

It is shown that the energy $(\varepsilon)$ and momentum $(k)$ dependences of the electron self-energy function $ Σ(k, \varepsilon + i0) \equiv Σ^{R}(k, \varepsilon) $ are, $ {\rm Im} Σ^{R} (k, \varepsilon) = -a\varepsilon^{2}|\varepsilon - ξ_{k}|^{- γ(k)} $ where $a$ is some constant, $ξ_{k} = \varepsilon(k)-μ, \varepsilon(k)$ being the band energy, and the critical exponent $ γ(k) $, which depends on the curvature of the Fermi surface at $ k $, satisfies, $ 0 \leq γ(k) \leq 1 $. This leads to a new type of electron liquid, which is the Fermi liquid in the limit of $ \varepsilon, ξ_{k} \rightarrow 0 $ but for $ ξ_{k} \neq 0 $ has a split one-particle spectra as in the Tomonaga-Luttinger liquid.

DOI
10.1143/jpsj.63.3923

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It is shown that the energy (ε)(\varepsilon)(ε) and momentum (k)(k)(k) dependences of the electron self-energy function Σ(k,ε+i0)≡ΣR(k,ε)Σ(k, \varepsilon + i0) \equiv Σ^{R}(k, \varepsilon)Σ(k,ε+i0)≡ΣR(k,ε) are, ImΣR(k,ε)=−aε2∣ε−ξk∣−γ(k){\rm Im} Σ^{R} (k, \varepsilon) = -a\varepsilon^{2}|\varepsilon - ξ_{k}|^{- γ(k)}ImΣR(k,ε)=−aε2∣ε−ξk​∣−γ(k) where aaa is some constant, ξk=ε(k)−μ,ε(k)ξ_{k} = \varepsilon(k)-μ, \varepsilon(k)ξk​=ε(k)−μ,ε(k) being the band energy, and the critical exponent γ(k)γ(k)γ(k), which depends on the curvature of the Fermi surface at kkk, satisfies, 0≤γ(k)≤10 \leq γ(k) \leq 10≤γ(k)≤1. This leads to a new type of electron liquid, which is the Fermi liquid in the limit of ε,ξk→0\varepsilon, ξ_{k} \rightarrow 0ε,ξk​→0 but for ξk≠0ξ_{k} \neq 0ξk​=0 has a split one-particle spectra as in the Tomonaga-Luttinger liquid.
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.
An exactly soluble one-dimensional model of electrons interacting with order-parameter fluctuations associated with short-range order is considered. The energy and momentum dependence of the electronic self-energy and spectral function are calculated and found to exhibit non-Fermi-liquid features similar to that seen for the two-dimensional Hubbard model: a pseudogap, shadow bands, anomalies in the self-energy, and breakdown of the quasiparticle picture. Deviations from Fermi-liquid behavior are largest close to the Fermi surface and as the correlation length increases.
For pure fermionic systems, and in the absence of symmetry breaking (e.g. magnetism, superconductivity, …), two types of metallic phases are well established: (i) the “normal” Fermi liquid phase in d = 3 presented in Chapters 12 and 19 via the response and the Green functions techniques; and (ii) the “anomalous” Luttinger liquid phase in d = 1 originally formulated with the bosonization technique by which the fermionic operators are represented in terms of “bosonic” density operators (Luther and Peschel (1974); Haldane (1981)) with the exact solution (Mattis and Lieb (1965)) of the Tomonaga–Luttinger model (Tomonaga (1950); Luttinger (1963)) discussed inmany reviews and books (see for instance the book by Giamarchi (2004)). Here, to stress similarities and differences between the Fermi liquid and the Luttinger liquid, we will follow the alternative route of response and Green functions techniques (Dzyaloshinskii and Larkin (1973), Di Castro and Metzner (1991), Metzner et al. (1998)) and add a few comments on the renormalization group approach (Metzner and Di Castro (1993)). As we have seen in Chapters 12 and 19, the Fermi liquid is described asymptotically by the free low-lying single-particle excitations, i.e., the quasiparticles with a zero-kelvin discontinuous occupation number in momentum space n k (see Eq. (18.54)) at a well defined Fermi surface (k = k F ): where here k F> and k F< indicate the limit k → k F from above and below. This discontinuity still marks the Fermi surface in the interacting system. The finite reduction of the single-particle spectral weight z kF with respect to the Fermi gas (z kF = 1) is given by the finite (i.e. non-critical) “wavefunction renormalization” (see Eq. (18.94)). The presence of the discontinuity at the Fermi surface, together with the Pauli principle, compel the inverse quasiparticle lifetime to be τ −1 ≈ max( T 2 , ϵ 2 ), where _ is the deviation of the energy of the quasiparticle from the Fermi energy. The energy uncertainty ħτ −1 due to the finite lifetime of a quasiparticle near the Fermi surface is small compared to its energy ϵ , and the quasiparticle concept is well defined. All the momentum transferring scattering processes become asymptotically ineffective, as derived phenomenologically in Chapter 12 and confirmed microscopically in Chapter 19.
We study the one-dimensional $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}J$ model for generic couplings using two complementary theories, the extremely correlated Fermi liquid theory and time dependent density matrix renormalization group over a broad energy scale. The two methods provide a unique insight into the strong momentum dependence of the self-energy of this prototypical non-Fermi liquid, described at low energies as a Tomonaga-Luttinger liquid. We also demonstrate its intimate relationship to spin-charge separation, i.e., the splitting of Landau quasiparticles of higher dimensions into two constituents, driven by strong quantum fluctuations inherent in one dimension. The momentum distribution function, the spectral function, and the excitation dispersion of these two methods also compare well.
Luttinger theorem relates Fermi surface volume to electron density per spin nnn. It holds in an arbitrary Fermi liquid but is widely believed to break when a fermionic self-energy has a pole. We show that this is not necessarily the case. As an example, we consider a fermionic system consisting of two gapped bands, originating from a self-energy Σ(ω,k)Σ(ω, \mathbf{k})Σ(ω,k) with a pole, and a narrow residual quasiparticle band crossing ω=0ω=0ω=0, with a small residue Z≪1Z \ll 1Z≪1 (a residual Fermi liquid). We consider Model I, in which ZZZ also renormalizes quasiparticle bandwidth WWW to ZWZWZW and Model II, in which WWW remains unrenormalized. The Green's function for such a system G(ω,k)G(ω, \mathbf{k})G(ω,k) contains two zeros and three poles --- one from the quasiparticle band and two from the gapped bands. Yet, we show that in Model I, the Luttinger-Ward analysis is fully applicable for any Z>0Z >0Z>0, and the results are exactly the same as if the gapped bands did not exist. For Model II, the Luttinger-Ward analysis recovers once ZZZ exceeds some critical value, which depends on fermionic density.
The authors have calculated the electronic self-energy of the two-dimensional Hubbard model as a function of frequency and momentum for various band fillings. They present this up to second order in the interaction parameter and also up to infinite order for ladder diagrams in the two-particle scattering channel. The authors have carried out direct k-space integrations using a triangle method. For a half-filled perfectly nested band, the imaginary part, lm Sigma (k f , omega ), has a linear- omega dependence for nested Fermi surfaces which is consistent with the marginal Fermi liquid phenomenology for high-T c superconductors. At, and near, half-filling there is a strong low-energy peak in lm Sigma (2) (k, omega ) when k is off the Fermi surface. The physical origin of this peak is discussed. The peak is enhanced in the ladder approximation. The corresponding structure in Re Sigma (2) (k, omega ) leads to multiple solutions of the Dyson equation for a wide range of values of the parameters. This leads to an appealing explanation of observed structure in some experimental angle-resolved photoemission spectra of high-T c materials. The authors also find anti-bound states split off from the band in the two-particle scattering ladders consistent with the suggestion of Anderson.