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I'm not (yet :-) ) an expert on superconductors, but one term I keep hearing all the time is that of the symmetry of the gap, which can be s-wave, p-wave, d-wave etc.

What exactly is the symmetry this refers to? I guess "something" will have spatial symmetries similar to the spherical harmonics for l = 0, 1, 2 etc. but what is this "something"?

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up vote 7 down vote accepted

The something is the superconducting order parameter, which is loosely $\Delta_{\alpha\beta}(r-r')=\langle\psi_\alpha(r)\psi_\beta(r')\rangle$ where $\psi_{\uparrow(\downarrow)}$ is the operator that annihilates a spin up (spin down) electron. 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.

Except that $s, p, d$ and so forth label representations of $SO(3)$, whereas $\Delta$ should transform under the point group $G$ of the crystal, which is a discrete subgroup of $SO(3)$. So a single irreducible representation of $SO(3)$ like spin 2 (aka $d$) actually may break up into multiple irreps of $G$. So the usual language is somewhat confusing, but any lengthy paper should define precisely what it means.

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Would you please give us such a lengthy paper as a reference ? I've tried reading the review by Sigrist and Ueda, the book by Samokhin and Mineev, and there is no so clear explanation as you gave us in these references if I remember correctly (I was a young researcher at that time and I didn't try to reopen this question since few years.) Thanks anyway for your beautiful answer. –  FraSchelle Dec 9 '12 at 22:53

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