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arXiv:cond-mat/9612248 · Modern Physics Letters B

A CONSTRAINT ON THE ANOMALOUS GREEN’S FUNCTION

Yong‐Jihn Kim

Published May 20, 1996• 7 citations
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Abstract

It is shown that the physical constraint of the Anomalous Green’s function gives a natural pairing condition. The resulting self-consistency equation is directly related to the BCS gap equation. Both inhomogeneous and homogeneous systems are considered to illustrate the importance of the constraint. Especially we find weak localization correction to the phonon-mediated interaction.

DOI
10.1142/s0217984996000614

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It is shown that the physical constraint of the Anomalous Green’s function gives a natural pairing condition. The resulting self-consistency equation is directly related to the BCS gap equation. Both inhomogeneous and homogeneous systems are considered to illustrate the importance of the constraint. Especially we find weak localization correction to the phonon-mediated interaction.
It is shown that the Bogoliubov–de Gennes equations pair the electrons in states which are linear combinations of the normal states. Accordingly, the BCS-like reduction procedure is required to choose a correct pairing. For a homogeneous system, we point out that the kernel of the self-consistency equation derived from the Bogoliubov–de Gennes equations needs to be constrained by the BCS pairing condition. In the presence of ordinary impurities, on the other hand, the Bogoliubov–de Gennes equations should be supplemented by Anderson's pairing condition to obtain the correct vacuum state by the corresponding unitary transformation. This results in localization correction to the phonon-mediated interaction.
A general theory of superconductivity is formulated within the thermodynamic Green function method for various types of pairing mediated by phonons, spin fluctuations, and strong Coulomb correlations in the Hubbard and t-J models.A rigorous Dyson equation for matrix Green functions is derived in terms of a self-energy as a many-particle Green function.By applying the noncrossing approximation for the self-energy, a closed selfconsistent system of equations is obtained, similar to the conventional Eliashberg equations.A brief discussion of superconductivity mediated by kinematic interaction with an estimation of a superconducting transition temperature in the Hubbard model is given.
We introduce a scheme through which the ground-state electronic density n and total energy Ev [n] of inhomogeneous interacting systems can be determined in a self-consistent (SC) way. It is based on a SC perturbation expansion for the single-particle Green function G. Our employed expansion is argued to be valid under all conditions provided that the ground-state charge density of the interacting system is pure-state non-interacting v representable. The formalism discussed in this paper has been applied to a quasi-one-dimensional periodic system. It has even been possible to carry out SC calculations with the dynamically screened exchange diagram (the so-called GW diagram, where W is the dynamically screened electron-electron interaction function) for the self-energy operator Σ. A coherent mode of interpretation of a number of our results entails that the overwhelming success of the GW framework in relatively accurately reproducing the experimental energy gaps of a number of semiconductors and insulators may be a direct consequence of non-self-consistency effects; whereas a non-self-consistent calculation in our tests increases the value of the local-density approximation (LDA) gap in the range of 40–65% of the ‘experimental’ gaps, self-consistency effects reduce this range to 14–17%. The same line of argument leads us to conclude that, in our model, 26–35% of the LDA error in the gap energies is due to the absence of a well known discontinuity in the LDA exchange-correlation potential, and the remaining part is due to the Kohn-Sham eigenvalues not being identifiable with one-particle excitation energies. A natural conclusion that may be drawn from these observations is that a correct description of excitation energies (e.g. of the gap energy) cannot be achieved without including, at least approximately, the vertex part of Σ, which is entirely absent in the GW scheme.
Analytic expressions for the Bardeen–Cooper–Schrieffer (BCS) gap of a many-electron system with the BCS model interaction are obtained in one, two, and three dimensions in the weak coupling limit, but for arbitrary interaction width ν≡ℏωD/EF, 0<ν<∞. Here ℏωD is the maximum energy of a force-mediating boson, and EF is the Fermi energy (which is fixed by the electronic density). Our results address both phononic (ν≪1) as well as nonphononic (e.g., exciton, magnon, plasmon, etc.) pairing mechanisms where the mediating boson energies are not small compared with EF, provided weak electron–boson coupling prevails. The essential singularity in coupling, sometimes erroneously attributed to the two-dimensional character of the BCS model interaction with ν≪1, is shown to appear in one, two, and three dimensions before the limit ν→0 is taken.
A self-consistence equation describing a superconductor whose normal state is a Luttinger liquid is computed in a rigorous way by renormalization group techniques. The gap and the critical temperatures are quite different with respect to the ones of normal superconductors (i.e. superconductors whose normal state is a Fermi liquid) and in particular, the critical temperature can be quite high for suitable values of the parameters. Such self-consistence equation can be obtained in a BCS mean field theory for a model of two Mattis models exchanging Cooper pairs. The connection of this model with recent theories of high T c superconductivity is briefly discussed.