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arXiv:cond-mat/9808209

Pairing in the Three-Band Hubbard Model of the Cu-O Plane

Michele Cini, Gianluca Stefanucci, A. Balzarotti

Published Aug 19, 1998• 0 citations
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Abstract

By a canonical transformation of the three-band Hubbard model, we introduce an effective Hamiltonian for the propagation of two holes doped into the ground state of the Cu-O plane. When the pair belongs to the $^{1}B_{2}$ or $% ^{1}A_{2}$ Irreducible Representations of the C$_{4v}$ Group, the bare holes do not interact by the on-site repulsion; the effective interaction between the dressed holes is obtained analytically in terms of renormalized matrix elements, and generalizes earlier findings from cluster calculations. The Fermi liquid is unstable and numerical estimates with reasonable parameters of the binding energy of the pair are in the range of tens of meV. Our scheme naturally lends itself to embody phonon-mediated and other interactions which cannot occur in the Hubbard model but may give important contributions.

DOI
10.48550/arxiv.cond-mat/9808209

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By a canonical transformation of the three-band Hubbard model, we introduce an effective Hamiltonian for the propagation of two holes doped into the ground state of the Cu-O plane. When the pair belongs to the 1B2^{1}B_{2}1B2​ or % ^{1}A_{2} Irreducible Representations of the C4v_{4v}4v​ Group, the bare holes do not interact by the on-site repulsion; the effective interaction between the dressed holes is obtained analytically in terms of renormalized matrix elements, and generalizes earlier findings from cluster calculations. The Fermi liquid is unstable and numerical estimates with reasonable parameters of the binding energy of the pair are in the range of tens of meV. Our scheme naturally lends itself to embody phonon-mediated and other interactions which cannot occur in the Hubbard model but may give important contributions.
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.
We calculate the effective interaction 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: 1 B2 and 1 A2 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 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 Weff is known, the pair eigenfunction is determined by an integral equation. We present numerical estimates of the binding energy |∆ | of the pairs, which is in the physically interesting range of tens of meV if unscreened on-site repulsion parameters are used. 1 I.
The effective two-band $t\ensuremath{-}J$ Hamiltonian for holes in Cu-O planes of cuprates is obtained from the three-band Hubbard Hamiltonian. In this model a doped oxygen hole together with two neighboring copper holes is treated as a composite spin-12\frac{1}{2}21​ three-sites particle. The model is studied by exact diagonalization in a Cu16O32{\mathrm{Cu}}_{16}{\mathrm{O}}_{32}Cu16​O32​ square cluster. Both binding energy and hole-hole correlation function indicate that two holes bind together for J/t>0.11J/t>0.11J/t>0.11 and are predominantly located on the next-nearest oxygen sites in the same Cu-O chain. Electron-phonon interaction is studied in the adiabatic phonons approximation. Copper (\ensuremath{\pi},0) ``half-breathing'' mode is shown to couple strongly to doped holes and increase the hole-hole correlation as well as the binding energy. These results are compared with experimental data.
The fully symmetric Cu-O clusters with a central Cu have two-hole singlet eigenstates which are not affected by the on-site repulsion. These pairs become bound states if the number of holes in the cluster is appropriate; moreover, the quantization at half-integer values of a magnetic test flux is consistent with a superconducting pattern. To investigate the mechanism of pairing, we study the symmetric clusters with $<~$21 atoms and four holes by exact diagonalization. Further, by the diagrammatic expansion of the ground-state energy and of the scattering amplitude, we obtain the spin-flip diagrams for the effective interaction, which is attractive for singlets and repulsive for triplets. A criterion for pairing in cluster calculations is thereby obtained.
We propose a configuration interaction mechanism that leads to hole pairs with zero repulsion energy which are exact eigenstates of the two-band Hubbard Hamiltonian. The two-hole pairs have symmetry and arise from degenerate states at the Fermi level; we show that they must exist independently of the band filling. To discuss the possible relevance of these states to the problem of pairing in high- superconductors we solve five- and nine-site model clusters with four holes. The quasiparticles become dressed by the interaction with the background holes and get paired with a set of valence band parameters well established for high- cuprates. We predict a number of important experimental facts such as binding energies of the correct order of magnitude, the symmetry of the order parameter and the singlet - triplet energy separation of the Cooper pairs.
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.