p-Wave Superconductivity and d-Vector Representation

Since the mid-80s, new classes of superconductors have been discovered in which the origin of superconductivity cannot be attributed to the electron–ion interactions at the heart of conventional superconductivity. Most of these unconventional superconductors are strongly correlated electron systems, and identifying (or even more difficult, predicting) the precise superconducting state has been, and sometimes remains, an actual challenge. However, in most cases, it has been demonstrated that in these materials the spin state of the Cooper pairs is a singlet state, often associated with a ‘d-wave’ or ‘\(s +/-\)’ orbital state. For a few systems, a spin-triplet state is strongly suspected, like in superfluid \(^3\)He; this leads to a much more complex superconducting order parameter. This was long supposed to be the case for the d-electron system Sr\(_2\)RuO\(_4\), and is very likely realized in some uranium-based (f-electron) ‘heavy fermions’ like UPt\(_3\) (with multiple superconducting phases) or UGe\(_2\) (with coexisting ferromagnetic order). Beyond the interest for these materials, p-wave superconductivity is presently quite fashionable for its topological properties and the prediction that it could host Majorana-like low energy excitations, seen as a route towards robust (topologically protected) qubits. The aim of these notes is to make students and experimentalists more familiar with the d-vector representation used to describe p-wave (spin triplet) superconductivity. The interest of this formalism will be illustrated on some systems where p-wave superconductivity is the prime suspect.