Flow Equations for Electron-Phonon Interactions

Peter Lenz, Franz Wegner

Published Jan 1, 1996• 38 citations

Abstract

A recently [9] proposed method of continuous unitary transformations is used to eliminate the interaction between electrons and phonons. The differential equations for the couplings represent an infinitesimal formulation of a sequence of Fröhlichtransformations [5]. The two approaches are compared. Our result will turn out to be less singular than Fröhlich’s. Furthermore the interaction between electrons belonging to a Cooper-pair will always be attractive in our approach. Even in the case where Fröhlich’s transformation is not defined (Fröhlich actually excluded these regions from the transformation), we obtain an elimination of the electron-phonon interaction. This is due to a sufficiently slow change of the phonon energies as a function of the flow parameter. 1 New address: Max-Planck-Institut für Kolloid- und Grenzflächenforschung, Kantstraße 55,

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OpenAlex

Abstract

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Jan 1, 199812 citations

ω_p

is much smaller than the Fermi energy

E_F

to that of the small polarons when

ω_p/E_F\ge 1

. (2)The jump of momentum distribution of electron number

n_{\bf k}

at the Fermi surface goes to zero when the dimensionless coupling constant

λ

increases to the critical value

λ_c\ge 1

, which means a possible broken down of the Fermi-liquid description.

{D}_{\mathbf{k}}

is electron-phonon coupling, ${\ensuremath{\omega}}_{\mathbf{k}}$ is the phonon energy, and

{\mathbf{k}}_{F}

is the Fermi momentum. The form of higher-order terms of the original Fr\"ohlich-transformed ${H}_{e\text{\ensuremath{-}}\mathit{ph}}$ is also found, up to terms proportional to the sixth power of the coupling---that is, up to those which yield the effective four-electron interactions.