Multiple odd-frequency superconducting states in buckled quantum spin Hall insulators with time-reversal symmetry

We consider a buckled quantum spin Hall insulator (QSHI), such as silicene, proximity-coupled to a conventional spin-singlet, s-wave superconductor. Even limiting the discussion to the disorder-robust s-wave pairing symmetry, we find both odd-frequency ($\omega$), spin-singlet and spin-triplet pair amplitudes and where both preserve time-reversal symmetry. Our results show that there are two unrelated mechanisms generating these different odd-$\omega$ pair amplitudes. The spin-singlet state is due to the strong inter-orbital processes present in the QSHI. It is exists generically at the edges of the QSHI, but also in the bulk in heavily doped regime if an electric field is applied. The spin-triplet state requires a finite gradient in the proximity-induced superconducting order along the edge, which we find is automatically generated at the atomic scale for armchair edges but not at zigzag edges. In combination these results make superconducting QSHIs a very exciting venue for investigating not only the existence of odd-$\omega$ superconductivity, but also the interplay between different odd-$\omega$ states.