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I have read in many places that the gapless surface states of 3D topological insulators are robust to perturbations which do not break time-reversal symmetry.

I have recently also seen many papers (such as http://arxiv.org/abs/1002.0842) on topological insulators in proximity to a superconductor. They all predict that the surface states will get gapped and that superconductivity is induced at the surface.

But I am not able to reconcile the two issues. Superconductivity does not break time-reversal symmetry. Then why is a gap produced in the surface states?

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  • $\begingroup$ Well, superconductors are famously known to have a gap, no ? So by proximity effect they share this gap with the metal, or semiconductor in contact with them, that's all. $\endgroup$ – FraSchelle Jul 14 '14 at 16:47
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The BiSb type topological insulators are also protected by particle-number/charge conservation symmetry which a superconductor would break. http://arxiv.org/pdf/0901.2686v2.pdf

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If time-reversal symmetry is broken at the surface of a topological insulator, a gap could open at the Dirac point of the topological surface state. The Dirac point, where forward- and backward-moving electrons have the same energy, is located at a time-reversal invariant momentum point (also called a Kramer's point) in the reciprocal space (the crystal momentum is zero or zero plus an integer multiple of a reciprocal lattice vector). The Kramer's pair of surface state branches defined for forward- and backward-moving electrons, which would otherwise be degenerate at k = 0, are no longer degenerate as a result of a time-reversal symmetry breaking interaction which couples differently to different spins.

For a superconductor, the energy gap forms at the Fermi energy (binding energy = 0). The condensate of Cooper pairs has a lower energy than the Fermi energy of the normal metal state, while unpaired electron states exist above the Fermi energy. Hence, a gap forms. The hope is that Cooper pairs will tunnel from a superconductor onto the surface of a topological insulator in proximity to it, creating a similar gap in the topological surface state.

So, these are two distinct kinds of gaps in different regions of the band structure which signify very different kinds of order.

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