Analogue viscous current flow near the onset of superconductivity

This is a pioneer paper about hydrodynamics in superconductors with rather specific predications, which can be experimentally verified.

Neglect of real-world complications and in particular paraconductivity just above Tc.

The paper argues that in a 2D superconductor in a narrow temperature window above the BKT critical temperature and below the GL critical temperature, the conductivity should depend on wave-vector because of the viscosity due to the presence of short-lived vortices. The main result of the paper is equation 14, which implies that DC conductivity at finite wave-vector should be smaller than DC conductivity at k=0. The authors argue that this can be tested by measuring the electrical resistivity of a 2D superconductor in a constricted geometry.

I think this is a nice contribution to the ongoing quest for finding different contexts for the observation of viscous electronic flow. The theoretical prediction looks easy to verify. In Figure 2, the authors show what the temperature dependence of the resistance of such constricted superconductors should be. However, they do not compare them with the actual resistivity curves reported in ref. 32 and 47 on much wider samples. Now, samples with a thickness of ~nm and a width of ~mm, show rounded transitions above Tc because of the well-known contribution of short-lived Cooper pairs (This is known as Aslamazov-Larkin fluctuations) . This contribution, the famous paraconductivity, is prominent up to at least 10 percent of Tc. In contrast, the effect discussed in this paper is restricted to a much narrower temperature range (~0.001 Tc). Just by comparing the evolution of the three theoretical curves for 0.5, 2 and 10 microns, one can see that it is unlikely to find what is seen in ref. 47 for a 1000 micron sample (such as S197 in their Table 1 and Figure 1).
All this suggests that the paper may be a nice academic contribution, but less directly connected to experimental reality than it is claimed.

An issue of terminology: "non-Ohmic" usually means current nonlinear in voltage (and vice versa). In this paper it appears that non-Ohmic is being used to mean linear but non-local. This is potentially misleading and should be stated much more clearly, since this is quite non-standard usage. I think it would be best to not use "non-Ohmic" to mean anything other than nonlinear, thus following standard usage. Non-local is probably a clearer and less ambiguous term in this context.

Since this paper is a discussion of systems near the Kosterlitz-Thouless transition, there should be a more careful and thorough discussion of the contributions of weakly bound vortex-antivortex pairs. The conductivity at k should, roughly, "see" weakly-bound pairs with spacings more than 1/k as effectively free vortices. Since the pairs interact logarithmically with distance, this should bring in power laws with continuously variable exponents as one varies the temperature, as is standard in Kosterlitz-Thouless physics. Ref. 20 works this out for uniform currents (k=0) as a function of the frequency. It seems that here one should do the analogous calculation, using the Kosterlitz-Thouless RG understanding of these systems, for zero frequency as a function of the wavenumber k. Perhaps an even simpler geometry to think about for theoretical purposes is a Corbino geometry, where a weak current (in linear response) is injected (from out of the plane) at a point (or a small patch) and flows in the plane out to infinity isotropically. There will be extra dissipation near the source, due to weakly bound vortex-antivortex pairs.