Odd-frequency superconducting pairing in multiband superconductors

We point out that essentially all multiband superconductors have an odd-frequency pairing component, as follows from a general symmetry analysis of even- and odd-frequency pairing states. We show that odd-frequency superconducting pairing requires only a finite band hybridization, or scattering, and nonidentical intraband order parameters, of which only one band needs to be superconducting. Under these conditions odd-frequency odd-interband pairing is always present. From a symmetry analysis we establish a complete reciprocity between parity in band index and frequency.

Figure 1

Odd-frequency odd-interband pair amplitude $F_{\omega }^o$ for a 3D two-band superconductor with ${\varepsilon }_a=-2{\sum }_i\cos \left(k_{i}a\right)$, ${\varepsilon }_b=\beta {\varepsilon }_a$ for $\beta =1$ (solid), $\beta =4$ (dashed), and ${\Delta }_a=0.5$, ${\Delta }_b=\alpha {\Delta }_a$ in Eq. (4). (a) $F_{\omega }^o$ as function of $\Gamma$ for $\alpha =0.5$, 0, $-1$ (increasing values). (b) $F_{\omega }^o$ as function of $\alpha$ for $\Gamma =0.1$, 0.5, 2 (increasing values).

Figure 2

Odd-frequency odd-interband pair amplitude $F_{\omega }^o$ in graphene for nearest-neighbor hopping $t=2.5$ eV, chemical potentials ${\mu }_{a,b}$, and on-site pairing potentials $U_{a,b}$. (a) $F_{\omega }^o$ as function of $U_b$ for $U_a=2t$ and ${\mu }_a={\mu }_b=0,0.25t,0.5t$ (increasing values). (b) $F_{\omega }^o$ as function of ${\mu }_b$ for ${\mu }_a=0$ (solid), $0.25t$ (dashed) for $U_a=U_b=2t,3t$ (increasing values). Zero $F_{\omega }^o$ for $\mu =0$ is caused by a superconducting quantum critical point at finite $U$.