Order parameter in the A phase of superfluid 3He

Up to now we considered isotropic superfluids, in which gauge invariance was broken but they remained invariant with respect to any three-dimensional rotation. In particular, in the Fermi superfluids the order parameter, or gap ∆, was a scalar independent of the direction. This means that the wave function of Cooper pairs was in the s state with zero orbital angular momentum and spin. Superconductors with such symmetry of the order parameter are called s-wave superconductors. In superfluid 3He the Cooper pair has a total spin and a total orbital moment equal to 1 (in unit ħ). Superconductors (charged superfluids), in which Cooper pairs have orbital momentum and spin equal to 1, are called spin-triplet or p-wave superconductors. In p-wave superfluids the order parameter is a 3 × 3 matrix with complex elements (18 parameters) in general (Vollhardt and Wölfle, 1990).

We focus our attention on the A phase of superfluid 3He, for which the order parameter matrix is a direct product of two three-dimensional vectors, which correspond to wave functions with spin 1 in the spin space and with orbital moment 1 in the orbital space. The unit vector d in the spin space determines the axis along which the spin of the Cooper pair exactly vanishes, although the spin modulus is equal to 1. Spin components along any other axis also vanish but only on average. So this spin wave function has no spin polarisation, and the state is analogous to the spin state in antiferromagnets with d being an analogue of the antiferromagnetic vector. In the orbital space there are two orthogonal unit vectors m and n, which determine a complex unit vector and a unit vector l = m × n. The vector l is called the orbital vector. It delineates the axis along which the orbital moment of the Cooper pair is directed. Neutral and charged superfluids with such an order parameter are called chiral or px + ipy superfluids. So the condensate of Cooper pairs has a spontaneous angular momentum along l, which is called an intrinsic angular momentum. In charged superfluids (px + ipy-wave superconductors) the intrinsic angular momentum leads to spontaneous magnetisation.