Nuclear Spin-Lattice Relaxation Rate in Odd-Frequency Superconductivity

We theoretically investigate the temperature dependence of nuclear spin-lattice relaxation rate $T_1^{-1}$ in bulk odd-frequency superconductivity. For a model odd-frequency pairing interaction, we first evaluate the superconducting order parameter, within the framework of the combined path-integral formalism with the saddle-point approximation. We then calculate $T_1^{-1}$ below the superconducting phase transition temperature $T_{\rm c}$, to see how the odd-frequency pairing affects this physical quantity. In the odd-frequency $p$-wave state, while the so-called coherence peak is suppressed as in the even-frequency $p$-wave case, $T_1^{-1}$ is found to exhibit the Korringa-law-like behavior ($T_1^{-1}\propto T$) except just below $T_{\rm c}$, even without impurity scatterings. In the odd-frequency $s$-wave case, the behavior of $T_{1}^{-1}$ is found to be sensitive to the detailed spin structure of the superconducting order parameter: In a case, $T_1^{-1}$ is enhanced far below $T_{\rm c}$, being in contrast to the conventional (even-frequency) $s$-wave BCS case, where the coherence peak appears just below $T_{\rm c}$. We also show that the calculated $T_1^{-1}$ in the odd-frequency $p$-wave case well explains the recent experiment on CeRh$_{0.5}$Ir$_{0.5}$In$_5$, where the possibility of odd-frequency $p$-wave superconductivity was recently suggested experimentally.