Phase-shifted Andreev levels in an altermagnet Josephson junction

We compute the effect of a d-wave magnetization (altermagnetism) on the spectrum of bound states (Andreev levels) in a junction between two s-wave superconductors (gap $\Delta_0$, phase difference $\phi$). Compared to a nonmagnetic junction, the $\phi$-dependence of the spectrum is shifted by an offset $\pm\delta\phi$, dependent on the spin direction, so that the Andreev levels become spin-polarized. In a planar junction, oriented along the crystalline axis of $d_{xy}$-wave symmetry, the excitation energies are determined by the normal-state transmission probability $T$ according to $E=\Delta_0\sqrt{1-T\sin^2\tfrac{1}{2}(\phi\pm\delta\phi)}$. We calculate the corresponding Josephson energy and supercurrent, recovering the 0-$\pi$ transition of related studies.