You misunderstood the classification I believe. Let's take an example.
In class D and 1D, the classification tells you there are two possible vacua (you understood this apparently). This is the famous $\mathbb{Z}_{2}$ ensemble in the classification. Next the classification tells you also that: at the boundary between the two gapped vacua, a Majorana mode emerges.
More precisely, an evanescent, localised mode emerges at the boundary between the two vacua, since they are gapped. For superconductors this emergent mode is a Majorana one, thanks to the particle-hole symmetry.
The construction is topological in essence: you map the problem of your differential equation to a group language, you recognise some properties (the Cartan class for instance, above the D one) which allow you to classify your problem (using equivalence relation, the more complicated part being to choose relevant criterion).Then these classes have additional properties, like the boundary term discussed above. To be a little bit more clear, it is topological because you want to understand how the local solutions to your problem are gluing to some other local ones in order to make global ones. Think about the Möbius stripe that is locally differentiable, but not globally (check out fiber bundle also, this is the mathematical object describing the property you're looking for). For the class D in 1D, you can find the two solutions in both vacua, but you can not glue them continuously without making a Majorana, continuity here means making a global continuous solution, or wave-function.
Nevertheless, the topological property tells you that something strange exists, it will never give you the wave-function. You can think about group classifications in quantum mechanics: the groups of your molecule/lattice-cell tell you which interaction terms exist, they never give you their strength (or the energy band splitting if you prefer to see the problem that way). A microscopic calculation is always required to get microscopic details. Now, there are some tricks: since the characteristic length inside a superconductor is the coherence length $\xi=\hbar v_{F}/\Delta$ ($\Delta$ gap parameter, $v_{F}$ Fermi velocity), you can think as a decaying wave-function as $\Psi_{\text{Maj.}}\sim e^{-x/\xi}$ at the $x=0$ interface. This is cheating because this is not a wave-function, but it nevertheless gives the correct estimate for the real wave function which should be something like $\Psi_{\text{Maj.}}\sim e^{-x/\xi}\sin k_{F}x$ for instance ($k_{F}$ Fermi wave-vector).
Next, some limitations of the classification:
- it works for non-interacting systems only (the Coulomb interaction is not taken into account for instance)
- it works for pure clean system only (sometimes this issue is not trivial: for (so-called topological) $p$-wave superconductor the superconductivity itself is destroyed by disorder; so perhaps the topological properties are conserved below the gap, but the gap vanishes due to disorder...)
so in practise it can never be applied to condensed matter problem. Some people infer the topological property to avoid discussing these points. I think these points are the most relevant one, though.
This post imported from StackExchange Physics at 2014-03-06 21:59 (UCT), posted by SE-user Oaoa
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