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Proceedings of the Estonian Academy of Sciences Physics Mathematics

DIFFERENCES BETWEEN ONE- AND MULTIBAND HUBBARD MODELS

G. Seibold, E. Sigmund

Published Feb 14, 1995• 0 citations
Publisher

Abstract

We investigate the large U limit of the one-and the three-band Hubbard model.It is found that the dynamics of a hole concerning the stabilization of spin fluctuations differs essentially in the two models.We therefore propose an alternative approach of mapping the three-band model onto an effective one-band Hamiltonian which describes the motion of an additional oxygen hole, in contrast to the usual mapping where the hopping of the Cu-O singlet is reduced to the motion ofthe Cu hole alone.

DOI
10.3176/phys.math.1995.2/3.20

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Physical review. B, Condensed matter•Sep 1, 1995•25 citations
We investigate the large U limit of the one-and the three-band Hubbard model.It is found that the dynamics of a hole concerning the stabilization of spin fluctuations differs essentially in the two models.We therefore propose an alternative approach of mapping the three-band model onto an effective one-band Hamiltonian which describes the motion of an additional oxygen hole, in contrast to the usual mapping where the hopping of the Cu-O singlet is reduced to the motion ofthe Cu hole alone.
A Hamiltonian for the motion of oxygen holes strongly coupled to the Cu spins in a one-dimensional (1D) Cu-O lattice is solved exactly for all dopings. This Hamiltonian retains the dominant O-hole hopping term, that is, spin-exchange hopping of the hole spin with the intervening Cu spin and neglects the weaker non-spin-exchange hopping term. The excitations naturally separate into two independent quasiparticles: spinons and holons. It is this Hamiltonian that should be taken as the correct zeroth-order Hamiltonian from which perturbation theory should be applied. We show that the spinon spectrum is the spectrum of the 1D antiferromagnet incommensurate with the Cu lattice and the holons are spinless noninteracting fermions in a cosine band. The Cu-Cu spin correlation in the ground state increases linearly with doping from antiferromagnetic (-0.443) at x=0 to ferromagnetic (\ensuremath{\approxeq}0.19) at x=1.
We calculate the properties of the two-band Hubbard model using the dynamical cluster approximation. The phase diagram resembles the generic phase diagram of the cuprates, showing a strong asymmetry with respect to electron- and hole-doped regimes, in agreement with experiment. Asymmetric features are also seen in one-particle spectral functions and in the charge, spin, and ddd-wave pairing-susceptibility functions. We address the possible reduction of the two-band model to a low-energy single-band one, as it was suggested by Zhang and Rice. Comparing the two-band Hubbard model properties with the single-band Hubbard model ones, we have found similar low-energy physics provided that the next-nearest-neighbor hopping term ${t}^{\ensuremath{'}}$ has a significant value $({t}^{\ensuremath{'}}∕t\ensuremath{\approx}0.3)$. The parameter ${t}^{\ensuremath{'}}$ is the main culprit for the electron-hole asymmetry. However, a significant value of ${t}^{\ensuremath{'}}$ cannot be provided in a strict Zhang and Rice [Phys. Rev. B 37, R3759 (1988); 41, 7243 (1990)] picture where the extra holes added into the system bind to the existing Cu holes forming local singlets. We note that by considering approximate singlet states, such as plaquette states, reasonable values of ${t}^{\ensuremath{'}}$, which capture qualitatively the physics of the two-band model, can be obtained. We conclude that a single-band $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}U$ Hubbard model captures the basic physics of the cuprates concerning superconductivity, antiferromagnetism, pseudogap, and electron-hole asymmetry, but is not suitable for a quantitative analysis or to describe physical properties involving energy scales larger than about 0.5 eV.
Using density matrix renormalization group calculations, we compare results obtained for the $t\ensuremath{-}J,$ one-band Hubbard, and three-band Hubbard models of a two-leg CuO ladder. Spin and charge gaps, pair binding energies, and effective pair hoppings are calculated for a wide range of parameters. All three models have an insulating state at a filling corresponding to one hole per Cu site. For physically relevant parameters their spin gaps are similar in size but they exhibit quite different charge gaps. We find that the binding energy of a pair of doped holes is of the order of the undoped ladder spin gap for all three models. The main difference between the models is the size of the effective pair hopping, which is significantly larger in the three-band model with parameters appropriate for CuO materials than in the other two models.
We compare numerical results of O and Cu occupations and optical and magnetic excitations for the three-band Hubbard model in a Cu4{\mathrm{Cu}}_{4}Cu4​O8{\mathrm{O}}_{8}O8​ cluster, with those obtained with an effective Hamiltonian Heff{\mathit{H}}_{\mathit{e}\mathit{f}\mathit{f}}Heff​ in which the Cu-O hopping tpd{\mathit{t}}_{\mathit{p}\mathit{d}}tpd​ is eliminated by means of a canonical transformation. This transformation up to order tpd4{\mathit{t}}_{\mathit{p}\mathit{d}}^{4}tpd4​ is not quantitatively valid when the relevant energy differences are of the order of \ensuremath{\surd}8 tpd{\mathit{t}}_{\mathit{p}\mathit{d}}tpd​ or less. However, an Heff{\mathit{H}}_{\mathit{e}\mathit{f}\mathit{f}}Heff​ with the form of the second-order one, but which retains all higher-order terms contained in a CuO4{\mathrm{CuO}}_{4}CuO4​ (Cu2{\mathrm{Cu}}_{2}Cu2​O7{\mathrm{O}}_{7}O7​) cluster for zero or one added hole (one added electron) to the undoped system, leads to a very good agreement with the results using the full Hilbert space. This Heff{\mathit{H}}_{\mathit{e}\mathit{f}\mathit{f}}Heff​ makes possible the numerical study of the three-band Hubbard model for realistic parameters, using larger clusters.
We perform a low-energy reduction of the three-band Hubbard Hamiltonian (H3b{\mathit{H}}_{3\mathit{b}}H3b​), keeping in the relevant Hilbert subspace not only local singlets (Zhang-Rice singlets), but also triplet states between Cu holes and O holes at the Wannier function of the same site, with x2{\mathit{x}}^{2}x2-y2{\mathit{y}}^{2}y2 symmetry. We solve exactly the resulting Hamiltonian HT{\mathit{H}}_{\mathit{T}}HT​ in a system of 2\ifmmode\times\else\texttimes\fi{}2 unit cells. From the analytical dependence of the parameters of HT{\mathit{H}}_{\mathit{T}}HT​ and the numerical results, one can see that the local triplet states can be practically neglected for finite O-Cu on-site energy difference \ensuremath{\Delta}, very large Cu on-site Coulomb repulsion Ud{\mathit{U}}_{\mathit{d}}Ud​, and O-O hopping tpp{\mathit{t}}_{\mathit{p}\mathit{p}}tpp​=0. This fact is in contrast with the mapping of H3b{\mathit{H}}_{3\mathit{b}}H3b​ to a one-band model using nonorthogonal singlets, which is very accurate when the Cu+{\mathrm{Cu}}^{+}Cu+ configuration can be neglected. Although the amount of local triplet states in the low-energy eigenstates is in general small, it increases with tpp{\mathit{t}}_{\mathit{p}\mathit{p}}tpp​ and for large tpp{\mathit{t}}_{\mathit{p}\mathit{p}}tpp​ it is necessary to introduce higher-order corrections in the one-band model to accurately represent the low-energy physics. In all cases even when local triplets are not important, the t-J model should be supplemented with other terms, to describe the lowest-energy levels. We also discuss briefly the effect of nonbonding O orbitals.