Gapless superconducting state and mirage gap in altermagnets

The interplay between the spin-orbit interaction (SOI) and magnetism produces interesting phenomena in superconductors. When a two-dimensional (2D) system with a strong SOI is coupled to an $s$-wave superconductor, an in-plane magnetic field can drive the system into a gapless superconducting state and induce a mirage gap at finite energies for an Ising superconductor. In this Letter, we demonstrate that when an $s$-wave superconductor is proximitized to an altermagnet, the intrinsic anisotropic spin splitting of the altermagnet can result in a gapless superconducting state and a pair of mirage gaps at finite energy. The gapless superconductivity exhibits spin-polarized segmented Fermi surfaces, with coexisting spin-singlet and spin-triplet pairings that have a $d$-wave character. Importantly, the gapless superconducting and mirage gap features are quantified through quantum transport. Our results suggest that the altermagnet is an ideal platform for studying gapless superconducting states and mirage gap physics.