Generalized dc and ac Josephson effects in antiferromagnets and in antiferromagnetic $d$-wave superconductors

The Josephson effect is generally described as Cooper-pair tunneling but it can also be understood in a more general context. The dc Josephson effect is the pseudo-Goldstone boson of two coupled systems with a broken continuous Abelian $U\left(1\right)$ symmetry. Hence, an analog should exist for systems with broken continuous non-Abelian symmetries. To exhibit the generality of the phenomenon and make predictions from a realistic model, we study tunneling between antiferromagnets and also between antiferromagnetic $d$-wave superconductors. Performing a calculation analogous to that of Ambegaokar and Baratoff for the Josephson junction, we find an equilibrium current of the staggered magnetization through the junction that, in antiferromagnets, is proportional to ${\hat{\mathbf{s}}}_L\times {\hat{\mathbf{s}}}_R$, where ${\hat{\mathbf{s}}}_L$ and ${\hat{\mathbf{s}}}_R$ are the Néel vectors on either sides of the junction. Microscopically, this effect exists because of the coherent tunneling of spin-one particle-hole pairs. In the presence of a magnetic field which is different on either sides of the junction, we find an analog of the ac Josephson effect where the angle between Néel vectors depends on time. In the case of antiferromagnetic $d$-wave superconductors we predict that there is a contribution to the critical current that depends on the antiferromagnetic order and a contribution to the spin-critical current that depends on superconducting order. The latter contributions come from tunneling of the triplet Cooper pair that is necessarily present in the ground state of an antiferromagnetic $d$-wave superconductor. All these effects appear to leading order in the square of the tunneling matrix elements.