1. Regarding the concept of singlet and triplet superconductor. In the very good answer to the question What is a $p_x + i p_y$ superconductor? Relation to topological superconductors the answerer says that

When there is no spin-orbit coupling, both the spin and the momentum are good quantum numbers (you need an infinite system for the second, but this is of no importance here), and one can separate $\Delta_{\alpha\beta}\left(\mathbf{k}\right)=\chi_{\alpha\beta}\Delta\left(\mathbf{k}\right)$ with $\chi_{\alpha \beta}$ a spinor matrix and $\Delta\left(\mathbf{k}\right)$ a function.

What is meant exactly in this context by "good quantum number"? Why the order parameter could not be an arbitrary mixture of triplet and singlet?

  1. Regarding the spatial symmetry of the order parameter. In this other answer (Pairing symmetry / superconducting gap symmetry) :

Now $\Delta$ must transform under the symmetry group of the crystal. So the terms $s, p, d$ and all their ilk refer to the possible representations.

I don't really get the first sentence. In this case, why the order parameter could not be an arbitrary mixture of these functions?

Maybe I can encompass both these questions by saying that in some papers, I have come across the idea that the order parameter should belong to one of the irreducible representations of the point group symmetry (of the lattice or the hamiltonian?). Why is it so?

Please don't hesitate to go in mathematical details or to suggest a comprehensive reference.


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