What is topological in Kitaev Chain? Realspace or the space of Pauli spins or the space of fermions?
My Understanding
I understand that majorana-zero modes are which are spatially separated, are protected in one phase. I am confused with the very notion of 'topology'! When talking about topology people show donut and coffee mug, which are topologically equivalent. What equivalence or inequivalence are we talking about in the context of the Kitaev chain?
On very general grounds, the idea behind topological order is to classify a system by means of a topological, and as such non-local, invariant. The latter describes the bulk of your phase without needing to consider the system’s boundaries. However, this rather abstract invariant becomes of physical meaning when it comes to interfaces between topologically trivial and topologically nontrivial phases (which you indeed distinguish by means of this topological invariant). That’s the bulk-boundary correspondence. When it comes to the Kitaev chain, there are several ways to define a topological bulk invariant, whose value reveals whether the first and the last Majorana Fermion of your chain (I’m assuming open boundary conditions) are left unpaired so to make for a zero-energy Fermionic mode. If you are wondering how this is related to the notion of topology made clear by the usual coffe-mug, donut and pretzel arguments, just think of the phase in which the topological invariant has one of two possible values as a sphere, a frisbee or whatever object without handles and of the other topologically nontrivial phase as a donut. As you can’t mould a frisbee into a donut without hollowing it, you can’t go from the non-trivial phase to the trivial phase (or viceversa) without going through a quantum phase transition, i.e. a critical point for which the energy spectrum of your many-body system becomes gapless.
