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Physical review. B, Condensed matter

Effective Hamiltonian for cuprate superconductors

C. D. Batista, Armando A. Aligia

Published Apr 1, 1993• 43 citations
Publisher

Abstract

We compare numerical results of O and Cu occupations and optical and magnetic excitations for the three-band Hubbard model in a ${\mathrm{Cu}}_{4}$${\mathrm{O}}_{8}$ cluster, with those obtained with an effective Hamiltonian ${\mathit{H}}_{\mathit{e}\mathit{f}\mathit{f}}$ in which the Cu-O hopping ${\mathit{t}}_{\mathit{p}\mathit{d}}$ is eliminated by means of a canonical transformation. This transformation up to order ${\mathit{t}}_{\mathit{p}\mathit{d}}^{4}$ is not quantitatively valid when the relevant energy differences are of the order of \ensuremath{\surd}8 ${\mathit{t}}_{\mathit{p}\mathit{d}}$ or less. However, an ${\mathit{H}}_{\mathit{e}\mathit{f}\mathit{f}}$ with the form of the second-order one, but which retains all higher-order terms contained in a ${\mathrm{CuO}}_{4}$ (${\mathrm{Cu}}_{2}$${\mathrm{O}}_{7}$) 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 ${\mathit{H}}_{\mathit{e}\mathit{f}\mathit{f}}$ makes possible the numerical study of the three-band Hubbard model for realistic parameters, using larger clusters.

DOI
10.1103/physrevb.47.8929

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We compare numerical results of O and Cu occupations and optical and magnetic excitations for the three-band Hubbard model in a {\mathrm{Cu}}_{4}$${\mathrm{O}}_{8} cluster, with those obtained with an effective Hamiltonian Heff{\mathit{H}}_{\mathit{e}\mathit{f}\mathit{f}}Heff​ in which the Cu-O hopping tpd{\mathit{t}}_{\mathit{p}\mathit{d}}tpd​ is eliminated by means of a canonical transformation. This transformation up to order tpd4{\mathit{t}}_{\mathit{p}\mathit{d}}^{4}tpd4​ is not quantitatively valid when the relevant energy differences are of the order of \ensuremath{\surd}8 tpd{\mathit{t}}_{\mathit{p}\mathit{d}}tpd​ or less. However, an Heff{\mathit{H}}_{\mathit{e}\mathit{f}\mathit{f}}Heff​ with the form of the second-order one, but which retains all higher-order terms contained in a CuO4{\mathrm{CuO}}_{4}CuO4​ ({\mathrm{Cu}}_{2}$${\mathrm{O}}_{7}) 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.
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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.
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We calculate the effective interaction WeffW_{eff}Weff​ between two holes added to the ground state of the repulsive three-band Hubbard model. To make contact with Cooper theory and with earlier Hubbard model cluster studies, we first use a perturbative canonical transformation, to generate a two-body Hamiltonian. Then, we extend the results to all orders. The approach is exact in principle, and we obtain a close analytic expression including explicitly the effects of all virtual transitions to 4-body intermediate states. Our scheme naturally lends itself to embody off-site, inter-planar, phonon-mediated and other interactions which are not considered in the Hubbard model but may well be important. The result depends qualitatively on the symmetry of the two-hole state: 1B2^{1}B_{2}1B2​ and $ ^{1}A_{2}$ pairs are special, because the bare holes do not interact by the on-site repulsion (W=0 pairs). The effective interaction in these channels is attractive and leads to a Cooper-like instability of the Fermi liquid; however WeffW_{eff}Weff​ is repulsive for Triplet pairs. Bound two-hole states of the same nature were reported earlier in small cluster calculations by exact diagonalisation methods; only symmetric clusters are good models of the plane. Once WeffW_{eff}Weff​ is known, the pair eigenfunction is determined by an integral equation. We present numerical estimates of the binding energy ∣Δ∣|Δ|∣Δ∣ of the pairs, which is in the physically interesting range of tens of meV if unscreened on-site repulsion parameters are used.
The so-called three-band Hubbard model is generally believed to be a good model for fermions on a CuO 2 lattice and to contain key features allowing to understand high-temperature superconductivity in the cuprates. A simpler and more popular model is the one-band Hubbard model. In this thesis,an extended one-band Hubbard model is derived from the three-band model. First, some mathematical background is given, as well as an introduction to Hubbard-like models. Then, to derive the extended one-band Hubbard model, the CuO 2 lattice is divided into clusters of one Cu and two O sites, and a variant of the Feshbach method is used to replace each such cluster by a single lattice site. To nd a suitable one-band model, all Hamiltonian matrix elements are matched between the two models. Finally, dierent sets of three-band parameters are considered, and it is studied how this aects the parameters in the extended one-band Hubbard model.