My understanding of the Kitaev honeycomb model is that high-level abstract properties (anyons and their braid statistics) can be seen to emerge the microscopics of the model (fermions and vortices). This then justifies the assumption that the effective theory of the model is Ising anyons.

Is the same true (evidence justifying an assumption) for the statement that the physics of edge of the system (for example on a disk) is described by a CFT? Or is there some deeper reason why the edge theory would have to be described by a CFT, even at the level of the microscopics?


1 Answer 1


I'm not sure this answers your question to the depth you would like, but let me give it a shot.

To properly set the scene: we are focusing on the so-called 'B phase' of Kitaev's paper, which is the spin liquid with emergent non-abelian anyons (defined by putting the ground state of $H = \sum S^\gamma S^\gamma$ in a magnetic field). In his paper, Kitaev essentially discusses three different ways of seeing that this has edge states described by the Ising CFT. Let me list them in order of ascending sophistication:

  1. Through a direct calculation in appendix B, he shows that the edge is described by a fermionic band that crosses the zero-energy line once. This means we have a single gapless $1+1d$ majorana on the edge. Moreover we see that it intersects the zero-energy line linearly, so it is a CFT. Indeed the action describing a single gapless majorana mode is known to be one of the equivalent formulations of the $1+1d$ Ising universality class. (In case these statements sound a bit mysterious to you, I can highly recommend John McGreevy's lecture notes on QFT) Note that this is the most down-to-earth way of seeing that the edge is described by a CFT, which moreover does not involve any assumptions. (Well, there is the assumption that the shape of the boundary is taken to be `sufficiently nice'.) The downside is perhaps that we are left wondering why the edge is described by this CFT, in which case (2) and (3) can help.
  2. In section 6.3, he calculates that the bulk has a non-zero Chern number $\nu = \pm 1$ (depending on the sign of the field). It is then `conventional wisdom' that a non-zero Chern number in the bulk corresponds to gapless edge modes on the boundary, with many arguments for this in the literature. See for example this stackexchange post. More conceptually, a non-zero Chern number implies the fermionic band has some non-trivial winding (around reciprocal space, which is shaped like a torus), and if this ever wants to connect to a trivial band (such as the outside of the system), the Chern number has to become ill-defined somewhere along the way (i.e. at the boundary). This means the band has to go through the zero-energy line, which generically defines a (linear, $1+1d$) gapless mode where that happens. Moreover since the Chern number $\nu$ counts the number of such non-trivial bands, the edge will have $\nu$ gapless fermions on the edge. The caveat here is that the Chern number is usually in the context of complex fermions. In Kitaev's case he is actually looking at a band of Majorana fermions, which you can think of as being half a usual fermion. So his $\nu = \pm 1$ then naturally implies that on the boundary we expect one gapless Majorana fermion. As dicussed in (1) above, this is the Ising CFT. (Note that Kitaev is far less sweeping in his statements than me (!), and in section 7 he discusses quite carefully how the non-zero Chern number for the Majorana bands implies the aforementioned edge modes.)
  3. The TQFT point of view. Firstly, let me point out that Kitaev in fact derives that the bulk is described by Ising anyons. Perhaps more clearly: he fully derives the anyon properties of the bulk, and any anyons with those properties are called Ising anyons. In fact, I'm not exactly sure what the nomenclature is, but I would say anything obeying the $\varepsilon \times \varepsilon = 1$, $\varepsilon \times \sigma = \sigma$ and $\sigma \times \sigma = 1 + \varepsilon$ fusion rules can be called Ising anyons, whereas Kitaev shows much more, deriving the topological spin and braiding rules. The question is then: to what extent does having bulk TQFT with Ising anyons imply that the boundary is described by an Ising CFT? Generally, if we have a $2+1d$ TQFT, we expect the boundary to be gapless (similar to the above intuition in (2)), and moreover under sufficiently nice conditions, in $1+1d$ a gapless system is usually equivalent to a CFT. So that's the 'expectation': I do not know of any real arguments that the boundary of a TQFT should be described by a CFT, but if anyone does, please let me know! But we can say the following: if the boundary of a TQFT is described by a CFT, the CFT is severely restricted by the TQFT. For example its OPE rules has to be consistent with the fusion rules of the bulk anyons. But time for more fun: in the case of the bulk Ising anyons of our above spin liquid, the CFT on the boundary is (pretty much) completely determined. This goes as follows: it turns out the quantum dimensions and topological spins of the bulk anyons determine the central charge of the boundary CFT up to mod $8$, which if we plug in the numbers relevant to our set-up, gives $c = \frac{1}{2} \mod 8$. So the `smallest' CFT that must necessarily live on the boundary has $c = \frac{1}{2}$, but there is only one CFT with this central charge: the Ising CFT!

I hope this helps.


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