Meissner effect of odd-frequency superconductors

Abstract

We present theoretical results on the Meissner effect of odd-frequency superconductors with the order parameter of the form, $\mathrm{sgn}({\ensuremath{\omega}}_{n})\ensuremath{\varphi}(k,i{\ensuremath{\omega}}_{n}),$ where $\ensuremath{\varphi}(k,i{\ensuremath{\omega}}_{n})$ is even in the Matsubara frequency ${\ensuremath{\omega}}_{n}$ as in the conventional case. It is shown that the spectral function of the anomalous Green's function is given in the form of the Hilbert transformation of the one for the even-frequency part, and the anomalous contribution to the paramagnetic kernel consists of the conventional term but with the opposite sign and a term expressed by the digamma function. In the static limit the latter term reduces to twice the former one with the opposite sign so that the net contribution becomes the same as the conventional one. This indicates the presence of the Meissner effect for this class of the odd-frequency superconductors. A model interaction is discussed that leads to the order parameter mentioned above.