Effect of the Vortices on the Nuclear Spin Relaxation Rate in the Unconventional Pairing States of the Organic Superconductor (TMTSF)${}_2$PF${}_6$

This Letter theoretically discusses quasiparticle states and nuclear spin relaxation rates $T_1^{-1}$ in a quasi-one-dimensional superconductor (TMTSF)${}_2$PF${}_6$ under a magnetic field applied parallel to the conduction chains. We study the effects of Josephson-type vortices on $T_1^{-1}$ by solving the Bogoliubov de Gennes equation for $p$-, $d$- or $f$-wave pairing interactions. In the presence of line nodes in pairing functions, $T_1^{-1}$ is proportional to $T$ in sufficiently low temperatures because quasiparticles induced by vortices at the Fermi energy relax spins. We also try to identify the pairing symmetry of (TMTSF)${}_2$PF${}_6$.