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Possible Odd-Frequency Superconductivity in Strong-Coupling Electron-Phonon Systems

A possibility of the odd-frequency pairing in the strong-coupling electron-phonon systems is discussed. Using the Holstein-Hubbard model, we demonstrate that the anomalously soft Einstein mode with the frequency $ω_{\rm E}\llω_{c}$ ($ω_{c}$ is the order of the renormalized bandwidth) mediates the s-wave odd-frequency triplet pairing against the ordinary even-frequency singlet pairing. It is necessary for the emergence of the odd-frequency pairing that the pairing interaction is strongly retarded as well as the strong coupling, since the pairing interaction for the odd-frequency pairing is effective only in the diagonal scattering channel, $(ω_{n},-ω_{n})\to(ω_{n'},-ω_{n'})$ with $ω_{n'}=ω_{n}\gtrsim ω_{\rm E}$. Namely, the odd-frequency superconductivity is realized in the opposite limit of the original BCS theory. The Ginzburg-Landau analysis in the strong-coupling region shows that the specific-heat discontinuity and the slope of the temperature dependence of the superfluid density can be quite small as compared with the BCS values, depending on the ratio of the transition temperature $T_{c}$ and $ω_{c}$.