Coexistence of Even- and Odd-Frequency Superconductivities Under Broken Time-Reversal Symmetry

Abstract

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.

Emergent Odd-Frequency Superconducting Order Parameter near Boundaries in Unconventional Superconductors

“…27,28 On the other hand, the relation in Eq. (2.10) does not hold anymore if the odd-frequency order parameter is the majority component.…”

“…24 As the attractive interaction of the spin-triplet s-wave, we consider an electron-phonon interaction as in our previous study under an external magnetic field. 27,28 We determine both the frequency and spatial dependences of the odd-frequency order parameter self-consistently on the basis of our previous quasi-classical theory for the surface and domain wall in the p-wave superconductors. 29 To understand the emergence of the odd-frequency order parameter, we express the induced odd-frequency pair amplitude with the superconducting quasi-particle wavefunction.…”

Strong-coupling superconductivity with mixed even- and odd-frequency pairing

“…In addition to this, the mass enhancement significantly suppresses the odd-frequency pairing, so that it is naturally expected that the induced odd-frequency component discussed below is also small as compared with the majority even-frequency component. 17 Performing the k integration, we obtain the free-energy functional as…”

“…The present authors proposed the mixing of the conventional s-wave singlet with the odd-frequency triplet component under applied uniform magnetic fields through Zeeman splitting of the conduction band. 17 Such a situation may also be relevant to superconductivity coexisting with ferromagnetism. 13,18 When the odd-frequency component is involved, a delicate treatment of the superconducting state should be required.…”

“…19,21 Nevertheless, we shall discuss in this paper that the above relation, + (k,iω n ) = (k,iω n ) * , is no longer satisfied simultaneously for both components in a coexistence case of the even-frequency and the odd-frequency pairings. 17 Consequently, the majority component of them takes the ordinary sign, + (k,iω n ) = + (k,iω n ) * , while the minority component must have the opposite sign, + (k,iω n ) = − (k,iω n ) * , in order to gain interference energy between two components. As a result, it exhibits different behaviors in various physical quantities from those obtained by neglecting this minority component with the anomalous negative sign relation.…”