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Physical review. B, Condensed matter

Cluster approach to the three-band Hubbard model of the Cu-O plane: Superconducting pairs

Michele Cini, A. Balzarotti

Published Dec 1, 1997• 12 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.56.14711

Related papers

Superconducting Pairs in Clusters and in the Cu-O Plane

Michele Cini, A. Balzarotti, Gianluca Stefanucci

Recently, by exact diagonalisation methods, we reported superconducting pairing and flux quantization in Cu-O Hubbard clusters of up to 21 atoms. The pairing is due to a configuration interaction mechanism driven by symmetry, and arises when the least bound holes have vanishing mutual repulsion ( W =0 pairs). The second-order, spin-flip interaction leads to a net attraction for the singlet and repulsion for the triplet. Here we show that electron bound pairs also exist and show similar flux quantization properties. Then, we extend the theory of pairing to the full plane and to all orders.

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Charge Conjugation and Pairing in a model Cu$_{5}$O$_{4}$ Cluster

Michele Cini, A. Balzarotti, Gianluca Stefanucci

Highly-symmetric three-band Hubbard Cu-O clusters have peculiar properties when the hole number is such that they admit W=0 hole pairs. These are two-hole eigenstates of the on-site Hubbard repulsion with eigenvalue 0, get bound by correlation effects when dressed by the interaction with the background, and cause superconducting flux quantization. We study the Cu$_{5}$O$_{4}$ cluster by exact diagonalization and show that bound electron pairs of $^{1}$% B$_{2}$ symmetry are obtained at an appropriate filling, and quantize flux like the hole pairs. The basic mechanism for pairing in this model is the second-order exchange diagram, and an approximate charge conjugation symmetry holds between electron and hole pairs. Further, the flux quantization property requires that the W=0 pairs of $d$ symmetry have $s$ symmetry couterparts, still with W=0; the former are due to a spin fluctuation, while the latter arise from a charge fluctuation mechanism. The simultaneous existence of both is an essential property of our model and is required for any model of superconducting $d$ pairs.

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Theory of pairing in the Cu-O Plane: Three-Band Hubbard Model and Beyond

Michele Cini, Gianluca Stefanucci, A. Balzarotti

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.

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Theory of pairing in the Cu-O plane: Three-band Hubbard model and beyond

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

Michele Cini, Gianluca Stefanucci, A. Balzarotti

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.

arXiv:cond-mat/9808209•Aug 19, 1998•0 citations
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.
Recently, by exact diagonalisation methods, we reported superconducting pairing and flux quantization in Cu-O Hubbard clusters of up to 21 atoms. The pairing is due to a configuration interaction mechanism driven by symmetry, and arises when the least bound holes have vanishing mutual repulsion ( W =0 pairs). The second-order, spin-flip interaction leads to a net attraction for the singlet and repulsion for the triplet. Here we show that electron bound pairs also exist and show similar flux quantization properties. Then, we extend the theory of pairing to the full plane and to all orders.
Charge Conjugation and Pairing in a model Cu5_{5}5​O4_{4}4​ Cluster
Highly-symmetric three-band Hubbard Cu-O clusters have peculiar properties when the hole number is such that they admit W=0 hole pairs. These are two-hole eigenstates of the on-site Hubbard repulsion with eigenvalue 0, get bound by correlation effects when dressed by the interaction with the background, and cause superconducting flux quantization. We study the Cu5_{5}5​O4_{4}4​ cluster by exact diagonalization and show that bound electron pairs of 1^{1}1% B2_{2}2​ symmetry are obtained at an appropriate filling, and quantize flux like the hole pairs. The basic mechanism for pairing in this model is the second-order exchange diagram, and an approximate charge conjugation symmetry holds between electron and hole pairs. Further, the flux quantization property requires that the W=0 pairs of ddd symmetry have sss symmetry couterparts, still with W=0; the former are due to a spin fluctuation, while the latter arise from a charge fluctuation mechanism. The simultaneous existence of both is an essential property of our model and is required for any model of superconducting ddd pairs.
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.
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.
In two-dimensional systems possessing a high degree of symmetry, the repulsive electron-electron interaction produces a pairing force; the mechanism would fail in the presence of strong distortions. We have studied this in the one-band and three-band Hubbard Model. From partially occupied orbitals one obtains pair eigenstates of the Hamiltonian with no on-site repulsion (the W=0 pairs). The concept of W=0 pairs allows to make qualitative and quantitative predictions about the behaviour of interacting many-body systems, a quite remarkable and unusual situation. Exact numerical solutions for clusters with ``magic'' hole numbers reveal attraction between the holes in W=0 pairs. The effect occurs in all fully symmetric clusters which are centered on a Cu site; then holes get paired in a wide, physically relevant parameter range and show superconducting quantization of the magnetic flux. A canonical transformation of the Hamiltonian, valid for clusters and for the full plane, leads to a Cooper-like equation for the W=0 pairs. We have evaluated the effective interaction and found that W=0 pairs are the bare quasiparticles which, once dressed, become two-hole bound states. We applied the above theory to the doped antiferromagnet, and found that the ground state at half filling is the singlet component of a determinantal state. We write down this determinant and the ground state wave function explicitly in terms of a many-body W=0 eigenstate. Our analytical results for the 4×44\times 44×4 square lattice at half filling and with doped holes, compared to available numerical data, demonstrate that the method, besides providing intuitive grasp on pairing, also has quantitative predictive power.
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.