Citation: Kitaev, Alexei. "Periodic table for topological insulators and superconductors." AIP conference proceedings. Vol. 1134. No. 1. American Institute of Physics, 2009.
Web: https://arxiv.org/abs/0901.2686
Tags: Foundational, Topological-insulators, Higher-dimensional, Fermionic-order
In this classic paper, Kitaev proposes a classification of free-fermion systems in arbitrary dimensions, in the case of no external symmetry, time-reversal (T) symmetry, and time reversal + charge conservation (T+Q) symmetry. The way this classification works is that one computes explicitly the homotopy type of the classifying space of gapped free-fermion Hamiltonians in a given dimension with a given generic symmetry. The point here is that free-fermion Hamiltonians are parameterized by gapped skew-Hermitian matrices. The condition of being gapped means that you can homotope the eigenvalues until they are all +1 or -1. You can then argue what moves on skew-Hermitian matrices will preserve the gap. For instance, if you have a block matrix with N +1 eigenvalues and N -1 eigenvalues and no off-diagonal terms, you can deform it to the identity matrix without closing the gap by an explicit homotopy. Importantly, the mathematical resulting from this structure is K-theory. This gives a 2-fold and 8-fold periodicity to the obtained results (Bott periodicity).
The classification is described in detail in several examples; in particular the d=1 no symmetries and the d=0 T+Q case. Especially in the d=0 case, the invariant can be understood using pure linear algebra - it is the sign of the Pffafian of the skew-symmetric matrix. While the classification is motivated and makes sense, it is not clear if it is correct in general from the arguments presented. Kitaev makes it clear that extra arguments are needed, and that he has figured out these arguments on his own. To quote his discussion on a necessary theorem: "That is the key technical result, but I cannot explain it in any detail in such a short note".
There is a similar paper to this inspired by Kitaev, whose discussion is complementary. It is worth reading for those wanting to understand the subject better:
> Ryu, Shinsei, et al. "Topological insulators and superconductors: tenfold way and dimensional hierarchy." New Journal of Physics 12.6 (2010): 065010.
Going beyond free-fermions, one expects that there should be a homotopy theory classification of interacting topological order. Some discussion is found in this MathOverflow post, as well as this paper quoting Kitaev's lectures:
> Gaiotto, Davide, and Theo Johnson-Freyd. "Symmetry protected topological phases and generalized cohomology." Journal of High Energy Physics 2019.5 (2019): 1-36.
There is also an interesting sort of grand unified theory inspired by Kitaev's work, also using K-theory:
> Schreiber, Urs, and Michael Shulman. "Quantum gauge field theory in cohesive homotopy type theory." arXiv preprint arXiv:1408.0054 (2014).
> Sati, Hisham, and Urs Schreiber. "Anyonic topological order in twisted equivariant differential (TED) K-theory." Reviews in Mathematical Physics 35.03 (2023): 2350001.