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Annalen der Physik

Derivation of Attractive Electron‐Electron Interactions for Superconductivity via Canonical Transformations

E. Kolley, Winfried Kolley

Published Jan 1, 1990• 2 citations
Publisher

Abstract

Abstract Two canonical transformations, related to the theory of conventional and hightemperature superconductivity are performed in detail. The BCS Hamiltonian is deduced from the Fröhlich Hamiltonian, and the t ‐ J model with attraction from the Hubbard model in the limit of large on‐site Coulomb repulsion. The underlying approximations are pointed out.

DOI
10.1002/andp.19905020209

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Abstract Two canonical transformations, related to the theory of conventional and hightemperature superconductivity are performed in detail. The BCS Hamiltonian is deduced from the Fröhlich Hamiltonian, and the t ‐ J model with attraction from the Hubbard model in the limit of large on‐site Coulomb repulsion. The underlying approximations are pointed out.
We extend previous studies of the (BCS) canonical approach for the attractive Hubbard model. A derivation of the BCS formulation is presented for both the Hubbard model and a simpler reduced Hamiltonian. Using direct diagonalization, exact one- and two-dimensional solutions for both Hamiltonians are compared to BCS variational calculations. Approximate and exact ground state energies and energy gaps are computed for different electron number systems as well as correlation observables not previously predicted. Reproducing published one-dimensional findings, the BCS approach is an excellent approximation for the Hubbard ground state energy but not the energy gap, a finding that remains true in two dimensions. Propagators and correlators are found more sensitive to wave functions and appreciable differences are computed with the Hubbard model exhibiting a weaker degree of superconductivity than the BCS. Even for weak coupling pronounced differences are found between the BCS and Hubbard on-site correlators. Significantly, in the extreme strong coupling limit the Hubbard model exhibits no dimensional crossover, or density induced transitions, to long-range order in two dimensions. However, for the reduced Hamiltonian the BCS approach is an excellent approximation for all observables in both one and two dimensions.
Abstract The method of canonical transformations proposed by one of the authors ten years ago in connection with a microscopic theory of superfluidity for Bose systems, is generalized here to Fermi systems, and forms the basis of a method for investigating the problem of superconductivity. Starting from Fröhlich's Hamiltonian, the energy of the superconducting ground state and the one‐Fermion and collective excitations corresponding to this state are obtained. It turns out that the final formulae for the ground state and one‐Fermion excitations recently obtained by Bardeen, Cooper and Schrieffer are correct in the first approximation. The physical picture appears to be closer to the one proposed by Schafroth, Butler and Blatt. The effect on superconductivity of the Coulomb interaction between the electrons is analyzed in detail. A criterion for the superfluidity of a Fermi system with a four‐line vertex Hamiltonian is established.
Motivation With few exceptions [131] it is widely accepted that the conventional Bardeen–Cooper–Schrieffer (BCS) theory [12] and its intermediate-coupling Eliashberg extension [14] do not suffice to explain high-temperature superconductivity. As shown in the preceding chapters the true origin of high-temperature superconductivity could be found in a proper combination of the bare Coulomb repulsion with the bare electron–phonon interaction in the framework of the many-body polaron theory [51] beyond the BCS–Eliashberg approximation. Nevertheless some researchers have maintained that the repulsive electron–electron interaction alone should provide pairing at high temperatures without phonons despite substantial experimental evidence that the EPI is an important player and needs to be included in the successful theory of high- T c superconductivity. Following the original proposal by P. W. Anderson, many authors [22, 74] argued that the electron–electron interaction in high-temperature superconductors was strong but repulsive providing high T c without phonons via superexchange and/or spin fluctuations in the d -wave pairing channel ( l = 2). A motivation for this concept can be found in the earlier works by Kohn and Luttinger (KL) [132, 133] who showed that the Cooper pairing of fermions with any weak repulsion was possible since the two-particle interaction induced by many-body effects is attractive for pairs with large orbital momenta, l ≫ 1.
We prove that the t-J-U model can be deduced from the Hubbard model at a large but finite U by a canonical transformation. We argue that the system may have a metal-insulator transition at a critical on-site Coulomb interaction whose value, however, is smaller than that in previous calculations in which the kinetic energy has a double counting. In a mean field theory and a special choice of the parameters, we show that the metallic state may be equivalent to the gossamer superconducting state proposed by Laughlin recently.
Superconductivity arises from a very complicated set of interactions between the electrons in a metal. This chapter considers a very simplified model of the interaction of a pair of electrons in a metal, it is an instructive example of how quantum mechanics works and the main features of this model are at the starting point of the modern theory of superconductivity. In a metal one has a repulsive Coulomb interaction between the electrons, together with an attractive interaction between the electrons and ions. The total interaction between any two electrons is a composite of this attraction plus the Coulomb repulsion. In some metals, this total interaction is attractive for electrons near the Fermi surface.
The method of canonical transformations is generalized to Fermi systems and forms the basis of a method for investigating the problem of superconductivity. Starting from Frohlich's Hanmiltonian, the energy of the superconducting ground state and the one-Fermion and collective excitations corresponding to this state are obtained. It turns out that the final formulas for the ground state and one-Fermion excitations recently obtained by Bardeen, Cooper, and Schrieffer are correct in the first approximation. The physical picture appears to be closer to the one proposed by Schafroth, Butler, and Blatt. The effect on superconductivity of the Coulomb interaction between the electrons is analyzed in detail. A criterion for the superfiuidity of a Fermi system with a four-line vertex Hamiltonian is established. (auth)