Time-reversal anomaly and Josephson effect in time-reversal-invariant topological superconductors

Topological superconductors are gapped superconductors with protected Majorana surface/edge states on the boundary. In this paper, we study the Josephson coupling between time-reversal-invariant topological superconductors and $s$-wave superconductors. The Majorana edge/surface states of time-reversal-invariant topological superconductors in all physical dimensions 1, 2, and 3 have a generic topological property which we name the time-reversal anomaly. Due to the time-reversal anomaly, the Josephson coupling prefers a nonzero phase difference between topological and trivial superconductors. The nontrivial Josephson coupling leads to a current-flux relation with a half period in a superconducting quantum interference device geometry, and also a half-period Fraunhofer effect in dimensions higher than 1. We also show that an in-plane magnetic field restores the ordinary Josephson coupling, as a sharp signature that the proposed effect is a consequence of the unique time-reversal property of the topological edge/surface states. Our proposal provides a simple and general approach to experimentally verify whether a time-reversal-invariant superconductor is topological.

Figure 1

(a) Schematic picture of a dc voltage-biased SQUID. (b) Dependence of the current oscillation amplitude on the flux $\Phi$. (c),(d) Numerical calculation of the dc SQUID $I/I_c$ as a function of $\varphi$ for the flux of $\Phi /{\Phi }_0=0,1/8,1/4,3/8,1/2$ in blue, purple, brown, green, and black respectively; (c) is for the slow limit and (d) for the fast limit with odd local fermion parity.

Figure 2

The Zeeman field effect on the Josephson current in the slow limit for $\delta t/\Delta =0.2$. (a) The Zeeman field is applied along the junction with the magnitude of $g{\mu }_{B}h/\Delta$ = 0 (black dashed), 0.005 (blue), 0.01 (red), and 0.02 (brown). (b) The Zeeman field of magnitude $g{\mu }_{B}h/\Delta =0.02$ applied along the junction (red) and perpendicular to the junction (blue). While the field along the junction changes the location of the discontinuity and eventually removes it altogether, the field perpendicular to the junction has negligible effects.

Figure 3

We have set $\left|\delta t\right|/\Delta =0.7$. (a) Numerical results for the current-phase relation $I\left(\varphi \right)$. (b) Numerical results for $I\left(\varphi \right)$ with the Zeeman field $\tilde{h}/\left|\Delta \right|=$ 0 (black), 0.02 (red), and 0.06 (blue) along the junction. (c) The scheme for the Fraunhofer diffraction experiment. $h_{\mathrm{out}}$ gives us the flux $\Phi$ through the junction, $h_{\mathrm{in}}$ gives rise to Eq. (11). (d) The Fraunhofer diffraction pattern with the Zeeman field $\tilde{h}/\left|\Delta \right|=$ 0 (black), 0.02 (red), and 0.06 (blue) along the junction; the filled curves are calculated from the current-phase relation Eq. (12) and the dots from the numerical calculation.

Figure 4

Plot of the Josephson current in $D=2$ with the spin-conserving hopping of $|\delta t/\Delta |=0.7$ and the spin-flip hopping of $|\lambda /\delta t|=0$ (blue dots) and $|\lambda /\delta t|=0.2$.