We calculate exact ground-state energies for small Hubbard-model clusters (molecules) and show that two electrons added to a molecule can experience an effective attraction despite their bare repulsive interaction (pair binding). This phenomenon takes place IonlyP for intermediate strength of the repulsive interactions, and on intermediate length scales. All the results are in qualitative agreement with those obtained using second-order perturbation theory up to surprisingly large values of the interactions and provide support for a purely electronic mechanism of superconductivity in doped ${\mathrm{C}}_{60}$.
We calculate exact ground-state energies for small Hubbard-model clusters (molecules) and show that two electrons added to a molecule can experience an effective attraction despite their bare repulsive interaction (pair binding). This phenomenon takes place IonlyP for intermediate strength of the repulsive interactions, and on intermediate length scales. All the results are in qualitative agreement with those obtained using second-order perturbation theory up to surprisingly large values of the interactions and provide support for a purely electronic mechanism of superconductivity in doped C60.
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: 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 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 ∣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.
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 4×4 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.
The electronic correlations on a C 20 molecule, as described by an extended Hubbard Hamiltonian with a nearest-neighbor Coulomb interaction of strength V , are studied using quantum Monte Carlo and exact diagonalization methods. For electron-doped C 20 , it is known that pair binding arising from a purely electronic mechanism is absent within the standard Hubbard model ( V = 0). Here we show that this is also the case for hole doping for 0< U / t ≤3 and that, for both electron and hole doping, the effect of a non-zero V is to work against pair binding. We also study the magnetic properties of the neutral molecule, and find transitions between spin singlet and triplet ground states for either fixed U or V values. In addition, spin, charge and pairing correlation functions on C 20 are computed. The spin–spin and charge–charge correlations are very short-range, although a weak enhancement in the pairing correlation is observed for a distance equal to the molecular diameter.
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.
Cluster perturbation theory is a technique for calculating the spectral weight of Hubbard models of strongly correlated electrons, which combines exact diagonalizations on small clusters with strong-coupling perturbation theory at leading order. It is exact in both the strong- and weak-coupling limits and provides a good approximation to the spectral function at any wave vector. Following the paper by S\'en\'echal et al. [Phys. Rev. Lett. 84, 522 (2000)], we provide a more complete description and derivation of the method. We illustrate some of its capabilities, in particular regarding the effect of doping, the calculation of ground state energy and double occupancy, the disappearance of the Fermi surface in the $t\ensuremath{-}{t}^{\ensuremath{'}}$ Hubbard model, and so on. The method is applicable to any model with on-site repulsion only.