# Why is there a Majorana zero mode in the $\pi$ flux core of the p+ip superconductor?

In this review paper (http://arxiv.org/pdf/1202.1293.pdf), the author shows that threading a $\pi$ flux through a 2D $p_x+ip_y$ superconductor will trap a Majorana zero mode at the flux center. The deduction involves solving differential equations, and is hard to generalize to arbitrary dimensions or arbitrary gauge defects. Since Majorana zero mode is topologically robust, I would expect a simpler and less detail-dependent way to show that there is indeed a Majorana zero mode in the $\pi$ flux of a $p_x+ip_y$ superconductor. Does anyone know such a simpler approach?

PS: The answer I expect should be general enough that can be applied to $\pi$-flux tubes in 3D topological insulator or topological superconductors as well.

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In my opinion (I don't found a paper that say the opposite) is there no generalization for arbitrary dimensions or arbitrary gauge defects. You must solve the Bogoliubov - de Gennes equation for each system. –  user27964 Mar 11 '14 at 9:47
The basic interpretation is that the gap parameter disappears at the vortex core. Then you ask yourself what is the degeneracy of the mode there. You then realise that $p_{x}+ip_{y}$ superconductivity does not has Kramers degeneracy. Then you conclude that a zero spinless mode exists: this is a Majorana mode since you have particle-hole symmetric theory from the beginning. –  FraSchelle Mar 12 '14 at 10:46
The classification should be clear enough to conclude the same thing. The vortex generates a non-trivial topology, that's it :-) The other non-trivial topology is the interface between two different vacua of the same Hamiltonian [more about that there physics.stackexchange.com/a/102079/16689 ] The microscopic calculation is complicated though, as usual with vortex, especially because quasi-classical theory is complicated. The interface problem is simpler to treat. The topological property of the quasi-classical theory might have something to do with Morse theory. –  FraSchelle Mar 12 '14 at 10:50