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In simplest terms, the presence of sub-gap zero energy localized modes (Majorana modes) makes a superconductor topological. A superconducting ground state is just a bunch of Cooper pairs and the BdG Hamiltonian describes excitations above the ground state. If the excitation spectrum has these localized modes then it is a topological superconductor otherwise ...

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The below seems to be a candidate for the first use of the term 'Majorana fermion'. (I'm not sure if it satisfies your other criteria.) Salam, Abdus, and J. Strathdee. Super-symmetry and non-Abelian gauges. International Centre for Theoretical Physics, Trieste (Italy), 1974.

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This is why people start to abandon calling the zero modes in vortices "Majorana fermions" because they are NOT fermions. $\gamma$ is a Majorana zero mode, which means it always has to pair up with another Majorana zero mode to form a 2-dimensional Hilbert space. Exchanging vortices generates a nontrivial unitary transformation on the degenerate space.

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There is no fundamental difference between the signatures found in the two works (Kouwenhoven and Yazdani). Both are tunneling spectroscopy, which roughly measures whether there is a zero-energy single-particle state in the spectrum. Yazdani's setup allowed him to do measurement away from the edge, so that the localization of the edge state can be directly ...

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