Retardation and strong-coupling effects in conventional superconductors

The electron-phonon interaction has been thoroughly studied for conventional superconductors. The much stronger Coulomb interaction has been given much less attention. Morel and Anderson (MA) showed that the effects on superconductivity are strongly reduced by retardation effects. MA studied the lowest diagram in a diagrammatic expansion. While the treatment of the electron-phonon interaction is greatly simplified by Migdal's theorem, there is no similar simplification for the Coulomb interaction U. The MA approximation is the leading contribution in U, but for large U an uncontrolled approximation. In a cooperation between the Metzner and Andersen departments, we have studied the second order diagram in U. To compare with exact dynamical mean-field theory (DMFT) results, we have studied the Holstein-Hubbard model in infinite dimensions.
By projecting out high-lying states down to the phonon energy, ωph, we can extract an effective Coulomb pseudopotential μ*, as a function of μ=U/D, shown by the red curve in the postscriptfile. Here D is half the band width. Although the diagram treated is of order U², the projection generates terms of all orders. The projection can be performed analytically to order U³, leading to
μ*=[μ+aμ²]/[ 1+μ log(D/ωph)+ aμ² log(α D/ωph)],
where a∼ 1 depends on details of the model. Neglecting the second order diagram (a=0), leads to the MA result, where the logarithm describes retardation effects. For the second order diagram, retardation effects are less efficient, as described by α=0.1, effectively reducing D. Nevertheless, as D → ∞, μ* → 0. The equation above is shown by the green curve in the figure. It agrees well with the numerical calculation (red curve) for μ ≤ 0.5. The MA result is also shown (blue curve). We have used the equation above to estimate the superconducting gap and compared with DMFT calculations. This suggest that diagrams up to second order in U are sufficient for μ ≤ 0.5. In conclusion, although MA is qualitatively correct, retardation effects are less efficient in second order.

We have also studied the validity of the Migdal-Eliashberg (ME) theory for conventional superconductors for large couplings λ. Although model calculations, based on the Hubbard-Holstein model have found that the ME theory breaks down for λ∼0.5, it is routinely applied for λ>1 to strong-coupling superconductors. To resolve this discrepancy, it is important to distinguish between bare parameters, used as input in models, and effective parameters, derived from experiments or density-functional calculations. Due to renormalization effects on the phonons, the effective parameters can be much larger than the bare ones. We use the DMFT formalism for a Hubbard-Holstein model in infinite dimension, where the DMFT is exact. Comparison of DMFT and ME results, shows that ME gives accurate results for the critical temperature and the spectral gap for large effective λ. This provides quantitative theoretical support for the applicability of the ME theory to strong-coupling conventional superconductors.

Publications:

Johannes Bauer, Jong E. Han, Olle Gunnarsson:
Corrections to the Coulomb pseudopotential in the theory of superconductivity,
arXiv1202.5051.

Johannes Bauer, Jong E. Han, and Olle Gunnarsson:
Quantitative reliability study of the Migdal-Eliashberg theory for strong electron-phonon coupling in superconductors
Phys. Rev. B 84, 184531 (2011).

For further information contact Olle Gunnarsson (gunnar@fkf.mpg.de).