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Physical review. B, Condensed matter

Quasiparticle bands of the two-dimensional Hubbard model

Yu Zhou, A. J. Fedro, S.P. Bowen, D. D. Koelling, T. C. Leung, B. N. Harmon, Suman Sinha

Published Nov 1, 1991• 5 citations
Publisher

Abstract

We present a mean-field calculation of the two-dimensional single-band Hubbard model based on a Mori-projection-operator formalism. Various physical properties including the energy, effective hopping matrix element, local moment, momentum distribution function, and Fermi surface are calculated. The results are compared with those of the Hubbard-I approximation, the four-boson mean-field theory of Kotliar and Rockenstein (SBMF), and quantum Monte Carlo simulations. Systematic improvement over the Hubbard-I approximation is seen, and good agreements are obtained between our calculation and the SBMF theory.

DOI
10.1103/physrevb.44.10291

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We present a mean-field calculation of the two-dimensional single-band Hubbard model based on a Mori-projection-operator formalism. Various physical properties including the energy, effective hopping matrix element, local moment, momentum distribution function, and Fermi surface are calculated. The results are compared with those of the Hubbard-I approximation, the four-boson mean-field theory of Kotliar and Rockenstein (SBMF), and quantum Monte Carlo simulations. Systematic improvement over the Hubbard-I approximation is seen, and good agreements are obtained between our calculation and the SBMF theory.
We present an exact formulation of the single-particle Green's functions for the two-dimensional three-band Hubbard model of the CuO2{\mathrm{CuO}}_{2}CuO2​ lattice based on a projection-operator formalism of Mori. The model is general and incorporates the various known local-density-approximation (LDA) interaction parameters. In this paper we present only the mean-field results. The inclusion of self-energy effects will be presented elsewhere. Already, in the mean-field approximation, we find that properties such as hole occupations and magnetic moments compare remarkably well with those from quantum Monte Carlo (MC) calculations. The densities of states are also calculated for various hole dopings, again using the parameters from the LDA calculations. Finally the detailed effect of hopping between different O p orbitals, are also presented.
Quantum Monte Carlo methods are used to study the physical properties of the two-dimensional Hubbard model for various band fillings. Results for the energy, magnetic moment, magnetic structure factor, and integrated optical spectral weight are presented. A comparison of the half-filled (〈n〉=1) and quarter-filled (〈n〉=0.5) bands is made using results for the momentum distribution 〈nk{n}_{k}nk​〉, the compressibility K, and the one-electron self-energy \ensuremath{\Sigma}(k,i${\ensuremath{\omega}}_{n}$), which shows that a gap exists in the one-electron excitations at half-filling but is absent for the one-quarter-filled case. At a filling of 〈n〉=0.87, the locus of k values for which 〈nk{n}_{k}nk​〉=0.5 is found to be close to the free Fermi surface for this filling.
Quasiparticle bands of the two-dimensional Hubbard model are calculated using the Roth two-pole approximation to the one-particle Green's function. Excellent agreement is obtained with recent Monte Carlo calculations, including an anomalous volume of the Fermi surface near half-filling, which can possibly be explained in terms of a breakdown of Fermi liquid theory. The calculated bands are very flat around the (\ensuremath{\pi},0) points of the Brillouin zone in agreement with photoemission measurements of cuprate superconductors. With doping there is a shift in spectral weight from the upper band to the lower band. The Roth method is extended to deal with superconductivity within a four-pole approximation allowing electron-hole mixing. It is shown that triplet p-wave pairing never occurs. A self-consistent solution with singlet ${\mathit{d}}_{{\mathit{x}}^{2}\mathrm{\ensuremath{-}}{\mathit{y}}^{2}}$-wave pairing is found and optimal doping occurs when the van Hove singularity, corresponding to the flat band part, lies at the Fermi level. Nearest-neighbor antiferromagnetic correlations play an important role in flattening the bands near the Fermi level and in favoring superconductivity. However, the mechanism for superconductivity is a local one, in contrast to spin-fluctuation exchange models. For reasonable values of the hopping parameter the transition temperature Tc{\mathit{T}}_{\mathit{c}}Tc​ is in the range 10--100 K. The optimum doping ${\mathrm{\ensuremath{\delta}}}_{\mathit{c}}$ lies between 0.14 and 0.25, depending on the ratio U/t. The gap equation has a BCS-like form and 2${\mathrm{\ensuremath{\Delta}}}_{\mathrm{max}}$/kTc{\mathit{kT}}_{\mathit{c}}kTc​\ensuremath{\simeq}4.
The one-band Hubbard model is considered in the slave-boson formulation first introduced by Kotliar and Ruckenstein. It is shown that a mean-field approximation, where broken-symmetry states appropriate for a bipartite lattice are allowed, leads to a quantitative agreement with quantum Monte Carlo results for local observables over a wide range of interactions (0U\ensuremath{\le}20t) and doping (0n〉\ensuremath{\le}1). Thus, our saddle-point solutions constitute an excellent starting point for a systematic treatment of fluctuations.
The canonical one-band Hubbard model is studied using a computational method that mixes the Monte Carlo procedure with the mean field approximation. This technique allows us to incorporate thermal fluctuations and the development of short-range magnetic order above ordering temperatures, contrary to the crude finite-temperature Hartree-Fock approximation, which incorrectly predicts a N\'eel temperature TN{T}_{N}TN​ that grows linearly with the Hubbard U/tU/tU/t. The effective model studied here contains quantum and classical degrees of freedom. It thus belongs to the ``spin fermion'' model family widely employed in other contexts. Using exact diagonalization, supplemented by the traveling cluster approximation, for the fermionic sector, and classical Monte Carlo for the classical fields, the Hubbard U/tU/tU/t vs temperature T/tT/tT/t phase diagram is studied employing large three- and two-dimensional clusters. We demonstrate that the method is capable of capturing the formation of local moments in the normal state without long-range order, the nonmonotonicity of TN{T}_{N}TN​ with increasing U/tU/tU/t, the development of gaps and pseudogaps in the density of states, and the two-peak structure in the specific heat. Extensive comparisons with determinant quantum Monte Carlo results suggest that the present approach is qualitatively, and often quantitatively, accurate, particularly at intermediate and high temperatures. Finally, we study the Hubbard model including plaquette diagonal hopping (i.e., the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}$ Hubbard model) in two dimensions and show that our approach allows us to study low-temperature properties where determinant quantum Monte Carlo fails due to the fermion sign problem. Future applications of this method include multiorbital Hubbard models such as those needed for iron-based superconductors.
Single-particle excitation spectra of the two-dimensional Hubbard model on the square lattice near half filling and at zero temperature are investigated on the basis of the self-consistent projection operator method. The method guarantees a high accuracy of the spectra with high energy and high momentum resolutions. It takes into account long-range intersite correlations as well as the strong on-site correlations. Effects of nonlocal excitations are clarified by comparing the results with those of the single-site approximation. The calculated spectra verify the quantum Monte-Carlo results for finite temperatures. The spectra at the Fermi level yield a hole-like Fermi surface in the underdoped region and an electron-like Fermi surface in the overdoped region. From a numerical analysis of the momentum dependent effective mass and self-energy, it is concluded that a marginal Fermi-liquid like state persists even at finite doping concentrations in the strongly correlated region because a van Hove singularity is pinned to the Fermi surface. It is also found that a kink structure appears in the quasiparticle energy band in the same region. The kink is shown to be caused by a mixing between the quasiparticle band and an excitation band with strong short-range antiferromagnetic correlations. These results suggest an explanation for some of the unusual properties of the normal state in high-TcT_{\rm c}Tc​ cuprates.