I don't understand what is meant by "gapless superconductors". As far as I know, the superconducting gap is necessairy for both zero resistance and the Meissner effect, the two hallmarks of superconductivity. So, what does it mean to have gapless superconductivity and how does it work?
Thanks in advance, Greetings
Superconductivity is characterized by a charged order parameter such as $\langle\psi_\alpha \psi_\beta\rangle \sim \epsilon_{\alpha\beta}\Phi$, where $\psi_\alpha$ is a spin 1/2 fermion of charge $e$, and $\Phi$ carries charge $2e$. If $\psi_\alpha$ is coupled to a $U(1)$ gauge field we find a Higgs (Anderson-Higgs-Meissner) effect, and the photon acquires a "mass".
The existence of a gap is a separate question. We look at the fermion propagator $G_{\alpha\beta}(\omega,k)=\langle\psi_\alpha\psi^\dagger_\beta\rangle_{\omega,k}$, and ask if there are any gapless excitations, that is, if $G(\omega,k)$ has poles $\omega=\epsilon_k$ with $\epsilon_k\to 0$ for some $k$. In (s-wave) BCS theory the absence of gapless modes is an automatic consequence of the presence of an order parameter, but that does not have to be the case.
A trivial counter-example is an order parameter with higher spin/orbital angula momentum. This occurs in $^3He$ ($p$-wave, though neutral) and high $T_c$ compounds (believed to be $d$-wave). In these cases the order parameter has nodal points or surfaces.
What is usually meant by gapless superconductivity is a more complicated situation, $s$-wave superconductivity in the presence of impurities (for example magnetic impurities, or ferro-magnetic order). In this case there is a range of temperatures where the order parameter persists, but the gap vanishes. The standard references are chapter 8 of de Gennes, Superconductivity of Metals and Alloys, as well as chapter 21 of Abrikosov, Fundamentals of the Theory of Metals.
