Appearance of Odd-Frequency Superconductivity in a Relativistic Scenario

Odd-frequency superconductivity is an exotic pairing mode in superconducting systems in which the symmetry of the gap function is odd in energy. Here we show that an inherent odd-frequency mode emerges dynamically under application of a Lorentz transformation of the anomalous Green function with an anisotropic gap function. To see this, we consider a Dirac model with quartic potential and perform a mean-field analysis to obtain a relativistic Bogoliubov-de Gennes system. Solving the resulting Gor'kov equations yields expressions for relativistic normal and anomalous Green functions. The form of the relativistically invariant pairing term is chosen such that it reduces to BCS form in the non-relativistic limit. We choose an ansatz for the gap function in a particular frame which is even-frequency and analyze the effects on the anomalous Green function under a boost into a relativistic frame. In the boosted frame the order parameter contains terms which are both even and odd in frequency: Δ(ω,k) ~ ω² → γ²ω² + β²γ²k² + 2βγ²kω. The relativistic correction to the anomalous Green function to first order in the boost parameter is completely odd in frequency. The odd-frequency pairing emerges dynamically as a result of the boost. This work provides evidence that odd-frequency pairing may form intrinsically within superconductors.