Inverse coherence effects in nuclear magnetic relaxation rates as a sign of topological superconductivity

We reveal that three-dimensional multi-orbital topological superconductivity can be identified by a bulk measurement, i.e., the temperature dependence of nuclear magnetic relaxation (NMR) rates. Below a critical temperature $T_{\rm c}$, the NMR rate in the topological state exhibits an anti-peak profile, which is opposite to the conventional $s$-wave state. This inversion coherence effect comes from a twist of order parameters with respect to orbital and spin degrees of freedom. Our self-consistent calculations in the model for Cu$_{x}$Bi$_{2}$Se$_{3}$ prove that the inverse coherence effect appears as a concave temperature dependence of the NMR rates. We propose that a time-reversal-invariant orbital-singlet spin-triplet topological superconductivity is characterized by the temperature dependence of the NMR rate.