BCS Theory (microscopic understanding) and Ginzburg-Landau (GL) Theory (phenomenological understanding) are two successful and complementary approaches to describe superconductivity. The way to derive analytically a GL-free energy functional from a path integral formulation of the BCS Theory, is, for conventional superconductivity, well known.
We propose to generalise this derivation for unconventional superconductivity. In conventional one, electrons are organised by Cooper pairs, that carry no orbital momentum (l = 0, s-wave) and, by symmetry, no spin (s = 0, spin triplet). Unconventional Cooper pairing, is defined as any state (l ≥ 0, s ∈ {0,1}). When l is odd (⇔ s = 1), the Cooper pair is in spin triplet configuration, allowing electrons to carry spin. In some context, existence of spin supercurrents can then be expected. Unconventional superconducting states are specified by the so-called d-vector representation, instead of the usual Δ-scalar gap function.
After finding a functional free energy (or, an effective action) generalised to unconventional pairings, our aim is to introduce SU(2) gauge in the microscopic action. It should allow for generation of spin supercurrents in the final functional, analogously to U(1) gauge fields that allow for charge supercurrents.
In addition, this path integral approach allows to find all the relevant coefficients of the power series in the functional, as a function of the material and external parameters. In particular, information about the chosen broken symmetry and so, the chosen superconducting state, can be deduced.