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 Δ₀ and phase difference φ). Compared to a nonmagnetic junction, the φ dependence of the spectrum is shifted by an offset ±δφ, dependent on the spin direction, so that the Andreev levels become spin polarized. In a planar junction, oriented along the crystalline axis of dxy-wave symmetry, the excitation energies are determined by the normal-state transmission probability T according to $E={{{Δ}}}₀√{1{-}T{{0.16em}{0ex}}{sin}²1/2({φ}±{}{δ}{φ}){{}⁰}}$. We calculate the corresponding Josephson energy and supercurrent, recovering the 0-${π}$ transition of related studies.