Optical and Raman selection rules for odd-parity clean superconductors

We derive selection rules in optical absorption and Raman scattering spectra that can determine the parity of pairing order parameters under inversion symmetry in two classes of clean superconductors: (1) chiral superconductors with strong spin-orbit couplings and (ii) singlet superconductors with negligible spin-orbit couplings. Experimentally, the inversion parity of pair wave functions can be determined by comparing the ``optical gap'' Δₒₚ in Raman and optical spectroscopy and the ``thermodynamic gap'' 2Δ in specific heat measurements, and the selection rules apply when Δₒₚ>2Δ. We demonstrate the selection rules for superconductivity in models of (1) doped Weyl semimetals and (2) doped graphene. Our derivation is based on the relation between pairing symmetry and the fermion projective symmetry group of a superconductor. We further derive similar selection rules for two-dimensional superconductors with twofold rotational symmetry and discuss how they apply to the superconducting state in magic-angle twisted bilayer graphene.