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arXiv:0902.1924

On the existence of Cooper pairs in the Hubbard Model

Valentin Voroshilov

Published Feb 11, 2009• 0 citations
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Abstract

By means of a new canonical transformation for the one-band Hubbard model at half filling we show the existence of Cooper pairs formed by strongly interacting quasiparticles.
DOI
10.48550/arxiv.0902.1924

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We have recently developed a canonical transformation of the Hubbard and related models, valid for systems of arbitrary size and for the full plane; this is particularly suited to study hole pairing. In this work we show that exact diagonalization results of the one band Hubbard model for small clusters with periodic boundary conditions agree well with the analytical ones obtained by means of our canonical transformation. In the presence of a pairing instability, the analytic approach allows us to identify the Cooper pairs. They are W = 0 pairs, that is, singlet two-hole eigenstates of the Hubbard Hamiltonian with vanishing on-site repulsion. Indeed, we find that the Coulomb interaction effects on W = 0 pairs are dynamically small, and repulsive or attractive, depending on the filling.
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.
We report on a numerical study of the two-dimensional Hubbard model and describe two new algorithms for the simulation of many-electron systems. These algorithms allow one to carry out simulations within the grand canonical ensemble at significantly lower temperatures than had previously been obtained and to calculate ground-state properties with fixed numbers of electrons. We present results for the two-dimensional Hubbard model with half- and quarter-filled bands. Our results support the existence of long-range antiferromagnetic order in the ground state at half-filling and its absence at quarter-filling. Results for the magnetic susceptibility and the momentum occupation along with an upper bound to the spin-wave spectrum are given. We find evidence for an attractive effective d-wave pairing interaction near half-filling but have not found evidence for a phase transition to a superconducting state.