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arXiv:cond-mat/0208482 · Journal of Physics Condensed Matter

Spin-disentangled exact diagonalization of repulsive Hubbard systems: superconducting pair propagation

Michele Cini, Gianluca Stefanucci, Enrico Perfetto, Agnese Callegari

Published Oct 18, 2002• 2 citations
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Abstract

By means of a novel exact-diagonalization technique, we show that bound pairs propagate between repulsive Hubbard clusters in a superconducting fashion. The size of the matrices that must be handled depends on the number of fermion configurations per spin , which is of the order of the square root of the overall size of the Hilbert space. We use CuO 4 units connected by weak O–O links to model the interplanar coupling and c -axis superconductivity in cuprates. The numerical evidence on Cu 2 O 8 and Cu 3 O 12 prompts the proposal of a new analytic scheme describing the propagation of bound pairs and also the superconducting flux quantization in a 3D geometry.

DOI
10.1088/0953-8984/14/43/103

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By means of a novel exact-diagonalization technique, we show that bound pairs propagate between repulsive Hubbard clusters in a superconducting fashion. The size of the matrices that must be handled depends on the number of fermion configurations per spin , which is of the order of the square root of the overall size of the Hilbert space. We use CuO 4 units connected by weak O–O links to model the interplanar coupling and c -axis superconductivity in cuprates. The numerical evidence on Cu 2 O 8 and Cu 3 O 12 prompts the proposal of a new analytic scheme describing the propagation of bound pairs and also the superconducting flux quantization in a 3D geometry.
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.
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 test the response to a threading magnetic field of rings of five-site C4v{C}_{4v}C4v​-symmetric repulsive Hubbard clusters connected by weak intercell links; each five-site unit has the topology of a CuO4{\mathrm{CuO}}_{4}CuO4​ cluster and a repulsive interaction is included on every site. In a numerical study of the three-unit ring with eight particles, we take advantage of an exact-diagonalization technique which can be generally applied to many-fermion problems. For O-O hopping we find a superconducting flux quantization (SFQ), but for purely Cu-Cu links bound pair propagation is hindered by symmetry. The results agree with W=0W=0W=0 pairing theory.
Lattice Hamiltonians with on-site interaction WWW have W=0 solutions, that is, many-body {\em singlet} eigenstates without double occupation. In particular, W=0 pairs give a clue to understand the pairing force in repulsive Hubbard models. These eigenstates are found in systems with high enough symmetry, like the square, hexagonal or triangular lattices. By a general theorem, we propose a systematic way to construct all the W=0 pairs of a given Hamiltonian. We also introduce a canonical transformation to calculate the effective interaction between the particles of such pairs. In geometries appropriate for the CuO2_{2}2​ planes of cuprate superconductors, armchair Carbon nanotubes or Cobalt Oxides planes, the dressed pair becomes a bound state in a physically relevant range of parameters. We also show that W=0 pairs quantize the magnetic flux like superconducting pairs do. The pairing mechanism breaks down in the presence of strong distortions. The W=0 pairs are also the building blocks for the antiferromagnetic ground state of the half-filled Hubbard model at weak coupling. Our analytical results for the 4×44\times 44×4 Hubbard square lattice, compared to available numerical data, demonstrate that the method, besides providing intuitive grasp on pairing, also has quantitative predictive power. We also consider including phonon effects in this scenario. Preliminary calculations with small clusters indicate that vector phonons hinder pairing while half-breathing modes are synergic with the W=0 pairing mechanism both at weak coupling and in the polaronic regime.
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.
A one-dimensional model of electrons locally coupled to spin-1/2 degrees of freedom is studied by numerical techniques. The model is one in the class of dynamic Hubbard models that describe the relaxation of an atomic orbital upon double-electron occupancy due to electron-electron interactions. We study the parameter regime where pairing occurs in this model by exact diagonalization of small clusters. World-line quantum Monte Carlo simulations support the results of exact diagonalization for larger systems and show that the kinetic energy is lowered when pairing occurs. The qualitative physics of this model and others in its class, obtained through approximate analytic calculations, is that superconductivity occurs through hole undressing even in parameter regimes where the effective on-site interaction is strongly repulsive. Our numerical results confirm the expected qualitative behavior and show that pairing will occur in a substantially larger parameter regime than predicted by the approximate low-energy effective Hamiltonian.