/sci/ - Science & Math

>>8748260
Topological superconductivity is superconductivity characterized by topological invariants, such as the Chern number. In general superconductivity arises due to closures of the energy band gap, and for many-body second quantized Hamiltonains with specific symmetries (such as P, C, T or combinations thereof) the topology of the Brillouin zone becomes important when characterizing these possible band gap opening/closures. For instance for a 2-dimensional Hamiltonian with PT symmetry, [math]n[/math] bands above and [math]m[/math] bands below the gap, the Chern numbers are given by the elements of the homotopy group [math]c^1 = -\frac{1}{4\pi}\int_{BZ}d^2k \operatorname{tr}\left(gdg^{-1}\right)^3 \in \pi_2(BZ) = \{BZ,S^2\}[/math], which correspond to the Hall conductances (in units of the flux quantum) across the system at zero temperature. In addition, these topological excitations are protected from the bulk in the sense that the edge modes can remain topologically nontrivial while the bulk transitions to a topologically trivial state.
Studying how these topological excitations braid and fuse with each other can tell you about the topological orders that exist in the system, and this is where category theory becomes useful. This can be used to characterize all possible topological materials in the world.