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arXiv:cond-mat/9701102 · International Journal of Modern Physics B

Pairing in the Bogoliubov–de Gennes Equations

Yong‐Jihn Kim

Published Jun 10, 1997• 5 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1142/s0217979297000885

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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.
The theory of superconductivity is formulated starting from the configuration-space form of the Bardeen-Cooper-Schrieffer (BCS) wave function. Self-consistent equations for treating nonu- niform systems are obtained. These equations have the advantage over the Bogoliubov-de Gennes equations that an effective condensate wave function and quasiparticle eigenstates can separately be obtained. The intrinsic orbital angular momentum density of the condensate is naturally iden- tified in this formalism due to a manifest two-particle bound-state nature in the one-particle density matrix.
The temperature dependence of an isolated quantum vortex, embedded in an otherwise homogeneous fermionic superfluid of infinite extent, is determined via the Bogoliubov--de Gennes (BdG) equations across the BCS-BEC crossover. Emphasis is given to the BCS side of this crossover, where it is physically relevant to extend this study up to the critical temperature for the loss of the superfluid phase, such that the size of the vortex increases without bound. To this end, two techniques are introduced. The first one solves the BdG equations with ``free boundary conditions,'' which allows one to determine with high accuracy how the vortex profile matches its asymptotic value at a large distance from the center, thus avoiding a common practice of constraining the vortex in a cylinder with infinite walls. The second one improves on the regularization procedure of the self-consistent gap equation when the interparticle interaction is of the contact type, and permits us to considerably reduce the time needed for its numerical integration by drawing elements from the derivation of the Gross-Pitaevskii equation for composite bosons starting from the BdG equations.
We investigate the mean-field ground state of a homogeneous three-component attractive Fermi gas at zero temperature. This is achieved by deriving a set of Bogoliubov--de Gennes (BdG) equations of the three-component system, including pairing both order parameters ${\ensuremath{\Delta}}_{\mathit{ij}}$ and coherence order parameters $\ensuremath{\langle}{\ensuremath{\psi}}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}{\ensuremath{\psi}}_{j}\ensuremath{\rangle}$ ($i\ensuremath{\ne}j$), where ${\ensuremath{\psi}}_{j}$ is the field operator for spin level jjj. Ward-Takahashi identities are obtained to constrain these order parameters. In addition, we present an explicit analytic mean-field solution for symmetric systems and verify that the quasiparticle excitations consist of both gapped and gapless spectra, which correspond to the excitations of paired and unpaired atoms. We further point out that the omission of $\ensuremath{\langle}{\ensuremath{\psi}}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}{\ensuremath{\psi}}_{j}\ensuremath{\rangle}$ in BdG equations could lead to an overestimation of ${\ensuremath{\Delta}}_{\mathit{ij}}$ in the strong coupling regime.