Meissner effect and topology

Question

Is there any analogy between Quantum Hall Effect and Meissner Effect? In other words, can we relate the existence of edge currents in superconductors (of type 1) with edge modes in QHE (that yield from the theorem that number of edge states should be equal to Chern number of the bulk states)?

Additionally, how appearance of Abrikosov vortices can be explained in terms of topology and Chern numbers?

Answer 1

This is one of those questions where I hope someone could give a positive answer, although I find it hard to imagine. Since I am not able to answer with a 'yes', let me instead give some reasons why they are dissimilar in some essential respects. For these reasons, I expect the answer to be 'no', but I will be very happy if someone can show the opposite.

1. Superconductors exist in any dimension, whereas the QHE is specific to even (spatial) dimensions. This is tied the fact that (the usual) Chern numbers are only defined in even dimensions.
2. The edge conductance of the QHE is discrete in nature, whereas you can have any current on the edge of a SC (of course up to a certain threshold)
3. The current at the edge of a Quantum Hall is completely determined by the Hamiltonian, whereas the current in a superconductor depends on its initial conditions/history.
4. You can't have a superconductor without breaking charge conservation but it can preserve time reversal symmetry. Compare this to the QHE which can only exist if time reversal symmetry is broken, yet it can preserve charge conservation. (There is the Quantum Spin Hall Effect which instead of breaking TRS is in fact protected by it, so this is still very different from the SC since the latter requires no symmetry protection.)
5. Related to (4), but still: the Meissner effect lives and breathes upon $\boldsymbol B = 0$, whereas the raison d'etre for QHE is $\boldsymbol B \neq 0$. (Of course I have to mention again that the QSHE is a counter-example to the latter statement though. But then I could say: the Meissner effect is based on (spontaneous) symmetry breaking, whereas the QSHE is based on symmetry protection.)
6. On a more physical note: the reason for the stability of the edge current in the QHE is due to the velocity at the edges (both magnitude and the direction) being fixed (by the Hamiltonian). This means that the only way to scatter is if it jumps from one edge to the the other, which is exponentially suppressed by the spatial separation of the edges. However, the stability in the case of the SC is intrinsically an emergent/many-body effect, where if one electron were to bump into something, due to quantum coherence many other electrons would have to simultaneously experience a similar bump, which is exponentially suppressed due to the number of particles involved which are spatially separated. (In other words, if just a single particle would bump, many electrons would notice and it hence would correspond to a huge energy cost.)

At least the last point can also be rephrased as a certain similarity: in both cases the stability is due to spatial separation. Still, it is quite different: the stability of QHE edge modes is due to the distance between the edges, and the stability of the SC edge current is due to the distance between the many correlated electrons. (For that same reason it is unlikely for one electron to escape out of a superconductor into a metal, whereas an electron in a Quantum Hall system has no problem doing so).