Constraints on superconducting pairing in altermagnets

Superconductivity in the recently discovered altermagnetic materials hosts large prospects for both fundamental physics and technological applications. In this work we show that a characteristic spin-sublattice locking in altermagnets puts severe constraints on possible superconducting pairing. In particular, we uncover that the most common form of superconductivity, uniform $s$-wave spin-singlet pairing is not possible to achieve in altermagnets. Considering an effective model for a $d_{x^2-y^2}$-wave altermagnet on a square lattice, we instead find that the most likely forms of spin-singlet pairing have $d_{x^2-y^2}$- or extended $s$-wave symmetry. We also find that the simplest form of equal-spin-triplet $p$-wave pairing is not allowed, but it can only exist as a mixed-spin-triplet $p$-wave state. We verify these constraints on pairing within an interaction-induced model of altermagnetism, where we also establish their validity for finite-momentum pairing. Additionally we discuss the possible pairing symmetries for odd-frequency superconducting pairing. Due to the generality of our results, they are applicable to both intrinsic superconductivity and proximity-induced superconductivity in altermagnet-superconductor hybrid junctions.