# How many of Kitaev's "Odds and Ends" in his 2006 anyon paper have been solved?

In Kitaev's 2006 paper Anyons in an exactly solved model and beyond, he lists nine open questions under the Section 10 "Odds and Ends". Briefly, these are

1. Find a condensed matter realization of the Kitaev model.
2. Use Abelian anyons for quantum memories.
3. Realize anyons in a lattice model with spontaneous time reversal symmetry breaking.
4. Create a stable four Majorana model in a Heisenberg antiferromagnet.
5. Extend the $$\nu \mod 16$$ classification of topological phases of $$\mathbb{Z}_2$$ gauge fields coupled to fermions to multilayer systems.
6. Prove or disprove that a topological phase is characterized by the chiral central charge $$c_-$$ and a modular tensory category.
7. Prove or disprove that CFTs are topological equivalent if they obey the conjecture in 6, ie there is a consistent 1D boundary between them.
8. A comprehensive study of BEC of the anyon vortices.
9. A mathematical framework for describing direct products of $$\nu$$ theories.

How many of these questions have been 'solved'?

1 has been partially solved in the following sense: Iridates amazingly have Kitaev interaction (among other interactions).

2 a lot has been known about this kind of phenomena, now part of the story of symmetry-enriched topological phase. Generally, defects of anyon-permuting symmetry must be non-Abelian and can be used to store quantum information. Ising-like defects in toric code were worked out in https://arxiv.org/abs/1004.1838.

3 has been solved by Yao and Kivelson: https://arxiv.org/abs/0708.0040

5 if the "multilayer" part is understood as some other global symmetry, then this is a problem of what's now called symmetry enriched topological phases and a great deal has been understood.

6 is wildly open if "topological phase" means some gapped phase on lattice. If topological phase is replaced by a topological quantum field theory, then I think it is settled. In fact now the classification of TQFTs has been done in 3+1 as well (works of Lan-Kong-Wen and Johnson-Freyd).

7 there has been many progress in establishing this conjecture (within some mathematical framework for CFT). The most recent one is https://arxiv.org/abs/1905.04924 and a follow up from Liang Kong and Hao Zheng.

8 and 9: the two go under the umbrella of "anyon condensation". Assuming that topological phases are described by MTC, then there is a by now well-developed theory of what it means to form a BEC of (bosonic) anyons (a good reference is https://arxiv.org/abs/1307.8244). In addition, the "E_8" state Kitaev mentioned in 8 is now understood as an example of invertible topological phase in 2+1.

• Thanks for such a comprehensive answer. Are 8 and 9 not particularly interesting to the community, or just less well posed? Jul 21 at 12:55
• @AndrewHardy I'm going to update the answer about 8 and 9. Jul 21 at 12:59