Competition between Triplet Superconductivity and Antiferromagnetism in Quasi One-Dimensional Electron Systems

We investigate the competition between antiferromagnetism and triplet superconductivity in quasi one-dimensional electron systems. We show that the two order parameters can be unified using a SO(4) symmetry and demonstrate the existence of such symmetry in one dimensional Luttinger liquids of interacting electrons. We argue that approximate SO(4) symmetry remains valid even when interchain hopping is strong enough to turn the system into a strongly anisotropic Fermi liquid. For unitary triplet superconductors SO(4) symmetry requires a first order transition between antiferromagnetic and superconducting phases. Analysis of thermal fluctuations shows that the transition between the normal and the superconducting phases is weakly first order, and the normal to antiferromagnet phase boundary has a tricritical point, with the transition being first order in the vicinity of the superconducting phase. We propose that this phase diagram explains coexistence regions between the superconducting and the antiferromagnetic phases, and between the antiferromagnetic and the normal phases observed in (TMTSF)$_2$PF$_6$. For non-unitary triplet superconductors the SO(4) symmetry predicts the existence of a mixed phase of antiferromagnetism and superconductivity. We discuss experimental tests of the SO(4) symmetry in neutron scattering and tunneling experiments.