Coexistence of spin density wave and triplet superconductivity in quasi-one-dimensional Bechgaard salts

The interplay between magnetic order and superconductivity is a very important problem in condensed matter physics. In the quasi-one-dimensional (quasi-1D) organic conductor (TMTSF)$_2$PF$_6$, an antiferromagnetic state charaterized by a spin density wave (SDW) order neighbors a triplet superconducting (TSC) state on the pressure-temperature phase diagram. Experiments [1,2] suggest a coexisting region of SDW and TSC orders in the vicinity of the phase boundary. We consider a tight-binding quasi-1D electron system, and construct the Ginzburg-Landau (GL) free energy with two order parameters. In the absence of a magnetic field, the rotational symmetry of this system is broken due to spin anisotropy and spin-orbit coupling. Thus, the GL free energy has a similar form as the ordinary $\phi_1$-$\phi_2$ model, except additional gradient terms. We calculate the GL coefficients microscopically and obtain a phase diagram in zero magnetic field. This phase diagram shows a coexistence region for SDW and TSC. \\ Reference:\\ $[1]$ T. Vuletic {\it et al.}, Eur. Phys. J. B {\bf 25}, 319 (2002).\\ $[2]$ I. J. Lee {\it et al.}, Phys. Rev. Lett. {\bf 94}, 197001 (2005).