Classification of pair symmetries in superconductors with unconventional magnetism

We consider unconventional magnets with proximity-induced superconductivity and investigate the emergence of superconducting correlations by carrying out a full classification of allowed Cooper pair symmetries. In particular, we focus on $d$-wave altermagnets and $p$-wave magnets under the influence of spin-singlet and spin-triplet superconductivity induced by proximity effect. Under generic conditions, we find that unconventional magnets not only drive a spin-singlet to spin-triplet conversion but also they transfer their parity symmetry that induces superconducting correlations with higher angular momentum. For instance, a conventional spin-singlet $s$-wave superconductor with $d$-wave altermagnetism is able to host odd-frequency mixed spin-triplet $d$-wave superconducting pair amplitudes, while when combining with $p$-wave magnetism the emerging superconducting pairing acquires an even-frequency mixed spin-triplet $p$-wave symmetry. We further demonstrate that unconventional magnetism produces even more exotic superconducting correlations in spin-singlet $d$-wave superconductors, where odd-frequency mixed spin-triplet $g$-wave and even-frequency mixed spin-triplet $f$-wave pair symmetries are possible in altermagnets and $p$-wave magnets, respectively. We also discuss how these ideas generalize to spin-triplet $p$-wave superconductors and show how our results can be applied to unconventional magnets with higher angular momentum, such as with $f$-, $g$-, and $i$-wave symmetries. Our results can help us understand the emergent superconducting correlations due to the interplay of unconventional magnetism and superconductivity.