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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Spin-dependent electric dipole operators are investigated group-theoretically for the emergence of an electric dipole induced by a single spin or by two spins, where the spin dependences are completely classified up to the quadratic order. For a single spin, a product of spin operators behaves as an even-parity electric quadrupole operator, which differs from an odd-parity electric dipole. The lack of the inversion symmetry allows the even-and odd-parity mixing, which leads to the electric dipole described by the electric quadruple operators. Point-group tables are given for classification of the possible spin-dependent electric dipoles and for the qualitative analysis of multiferroic properties, such as an emergent electric dipole moment coexisting with a magnetic moment, electromagnon excitation, and directional dichroism. The results can be applied to a magnetic ion in crystals or embedded in molecules at a site without the inversion symmetry. In the presence of an inversion symmetry, the electric dipole does not appear for a single spin. This is not the case for the electric dipole induced by two spins with antisymmetric spin dependence, which is known as vector spin chirality, in the presence of the inversion center between the two spins. In the absence of the inversion center, symmetric spin-dependent electric dipoles are also relevant. The detailed analysis of various symmetries of two-spin states is applied to spin dimer systems and the related multiferroic properties.

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