A novel superconducting state under the broken time-reversal symmetry is studied in conventional phonon-mediated superconductors. By solving the Eliashberg equation self-consistently with the mass renormalization effect, it is found that the even-and odd-frequency components of the order parameter coexist in the bulk system as a consequence of the broken time-reversal symmetry. This finding would direct more attention to the odd-frequency pairing that affects physical quantities, especially in strong coupling superconductors.

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A single impurity problem is investigated for multiband s-wave superconductors with different sign order parameters (s ± -wave superconductors) suggested in Fe-pnictide superconductors. Not only intraband but also interband scattering is considered at the impurity.The latter gives rise to impurity-induced local boundstates close to the impurity. We present an exact form of the energy of the local boundstates as a function of strength of the two types of impurity scattering. The essential role of the impurity is unchanged in finite number of impurities. The main conclusions for a single impurity problem help us understand effects of dense impurities in the s ± -wave superconductors. Local density of states around the single impurity is also investigated. We suggest impurity site nuclear magnetic resonance as a suitable experiment to probe the local boundstates that is peculiar to the s ± -wave state. We find that the s ± -wave model is mapped to a chiral d x 2 −y 2 ± id xy -wave, reflecting the unconventional nature of the sign reversing order parameter. For a quantum magnetic impurity, interband scattering destabilizes the Kondo singlet.is pair breaking for the s ± -wave superconductivity.For the symmetric scattering (U ++ = U +− ), one of the effective potential becomes zero, while the other is ±2U +− as in eq. (31). This means that only one of the conduction electron forming a Cooper pair is scattered by the potential, while the other is not. Using the effective Green's function reduced to the 2 × 2 matrix form, we can obtain the same boundstate energy E B defined in eq. (20). Classical magnetic scatteringLet us consider here magnetic scattering of Ising type. The matrixÛ in eq. (10) has the following form:

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