Does Superconductors offer zero resistance for AC current also? I was reading about Superconductivity in 8th edition of Introduction to Solid State Physics by Charles Kittel. The below given statement has confused me a bit:

'In the superconducting state, the DC electrical resistivity is zero.'

 A: At least type-II-superconductors allow magnetic flux tubes to enter because they have normal-conducting regions as well as superconducting regions (where magnetic flux gets expelled). This has the consequence that if the field (or current) changes, the flux tubes move, causing some resistive losses. It is a desired excercise of high-temperature superconduction to prevent the movement of the flux tubes by pinning them to lattice defects, in order to improve AC resistivity. The lower the frequency of the currents (approaching DC), the less need there is for flux-pinning. For DC, you finally get the ideal superconducting behavior Kittel is talking about.
See for example https://encyclopedia.pub/10222 or https://en.wikipedia.org/wiki/Flux_pinning.
A: It is easier to argue in terms of conductivity (the inverse of the resistivity). A classical, BCS superconductor, at 0K, has an infinite dc conductivity (zero resistance) but a vanishing ac conductivity.
A hand-waving over-simplified explanation comes from the Drude-model for free electrons. Its frequency dependent complex conductivity, $\sigma(\omega)$ is:
$$\sigma(\omega) = \frac{\sigma_0}{1 + i \omega \tau}$$
where $\sigma_0 = ne^2 \tau /m$ is the regular dc conductivity. $\tau$ is the average time between electron collisions that produce the resistivity. The real part of $\sigma(\omega)$, called $\sigma_1(\omega)$ is the material electrical conductivity, the inverse of the resistivity. It is:
$$\sigma_1(\omega) = \frac{\sigma_0}{1 + (\omega \tau)^2}$$
This is a lorentzian centered at zero-frequency (note that $\omega = 0$ means dc) and with a half-maximum width of $1/\tau$. The Drude model describes roughly a regular resistive metal.
Now let us think about a hypothetical perfect metal. This material would have no electron collisions, therefore $\tau \rightarrow \infty$ and, consequently, $1/\tau \rightarrow 0$. Using this in our $\sigma_1$ lorentzian yields $\sigma_1(\omega) \propto \delta(\omega)$. A perfect conductor would have an infinite dc conductivity but zero ac ($\omega > 0$) conductivity.
A superconductor, however, is not the same as a perfect conductor. There are a few differences, one of which is the existence of a superconducting gap energy ($2\Delta$). If one gives this energy to the material, it will destroy superconductivity. And this energy can be given through electromagnetic waves (a photon absorption in quantum mechanics lingo). So, once the photon energy ($\hbar \omega$) is above $2\Delta$, the superconducting properties for those fields disappear and the material comes back to its normal state.
The picture below sumarizes the whole thing.

In red you have the normal state Drude metal. In blue the superconducting state with a $\delta(\omega)$ dc conductivity, a zero ac conductivity up to photon energies of $2\Delta$ and a recovery to the normal metal response above. The rigourous calculation was done by Mattis and Bardeen (and that is what the blue curve shows).
Note that the picture above is only valid at T=0K. At finite temperature, there will be "thermally broken Cooper pairs" (regular electrons) that will give a residual, non zero, conductivity between $\omega = 0^{+}$ and $2 \Delta$. At low enough frequencies, at finite temperature, one can still have a large conductivity. At finite temperature the dc value is still infinite but this comes from the fact that residual electrons conductivity will have a $\omega \rightarrow 0$ divergence. The ac resistivity can be very small in that regime, but not zero.
So, yes, Kittel is right. The dc electrical resistivity is zero and, as the sentence implies, the ac is not zero. It can actually be pretty bad in some cases. For ac fields of low energy ($\hbar \omega \ll 2 \Delta$) one can still have a close to zero resistivity, but not zero as for the dc value.
A: There would be no AC electrical current resistance. There is no resistance in a superconductor, period. What the statement means is that you do not lose electrons to heat while transferring electrons through the superconductor. If you would like more information go to this website: https://cosmosmagazine.com/science/physics/resistance-is-futile-the-super-science-of-superconductors/ Also, please improve your question. Your question and body ask different questions.
