"Anyons in an exactly solved model and beyond", Alexei Kitaev, 2006

Reviewed March 23, 2024

Citation: Kitaev, Alexei. "Anyons in an exactly solved model and beyond." Annals of Physics 321.1 (2006): 2-111.

Web: https://arxiv.org/abs/cond-mat/0506438

Tags: Foundational, Majorana-fermions, Modular-tensor-categories

This paper is without a doubt one of the most foundational and information-packed in the entire theory of topological phases. The first line is wonderfully poetic: "Certainly, the main result of the paper is an exact solution of a particular two-dimensional quantum model. However, I was sitting on that result for too long, trying to perfect it, derive some properties of the model, and put them into a more general framework. Thus many ramifications have come along". I feel in much a similar way about this review. I've been aware of this paper and I had read parts of it even before I started keeping literature reviews. As I keep reading new parts of the paper and finding more details, I keep having more things to say. It'll probably take me years until I have fully digested this paper.

Here are some of the highlights, in my opinion:

This paper introduced the idea of modular tensor categories as anyons models. Certainly the connection between MTCs and topological phases was not unknown - MTCs were first defined in the context of modular functors, even. The new idea is to take all of these ideas outside of the context of field theory, and use them to describe simply the behavior of quasiparticle excitations in condensed matter systems. For most condensed matter theorists, this is the birth of categorical methods in condensed matter.
This paper introduces the honeycomb code. This is the first explicit Hamiltonian given for nonabelian anyons which seemed reasonable to appear as-is (or essentially as-is) in real 2D quantum material. This has since been essentially been all but confirmed, with the most prominent material being RuCl3. The Hamiltonian and its solution using Majorana degrees of freedom is seen as Kitaev's most important contribution to many working condensed matter theorists.
This paper introduced the sixteenfold way classification of topological order with onsite Z2 symmetry. This sixteenfold way (and its generalizations) are very important in the study of fermionic topological order, and the variety of deep insights given about Ising (and Ising-like) anyons in this paper have guided researchers working with these models.