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In 2D the entanglement entropy of a simply connected region goes like \begin{align} S_L \to \alpha L - \gamma + \cdots, \end{align} where $\gamma$ is the topological entanglement entropy.

$\gamma$ is apparently \begin{align} \gamma = \log \mathcal{D}, \end{align} where $\mathcal{D}$ is the total quantum dimension of the medium, given by \begin{align} \mathcal{D} = \sqrt{\sum_a d_a^2}, \end{align} and $d_a$ is the quantum dimension of a particle with charge $a$.

However, I do not quite understand what this quantum dimension is, or what a topological sector (with I guess charge $a$?) is. It is usually just quoted in papers, such as $\mathcal{D} = \sqrt{q}$ for the $1/q$ Laughlin state, $\gamma = \log 2$ for the Toric code... Could someone please explain? How do I know how many topological sectors a state (? system?) has, and how do I get its quantum dimension?

In addition, I am guessing that a topologically trivial state (i.e. not topological state) has $\mathcal{D} = 1$. Would that be right? What makes a state be non-trivial topologically (i.e. have $\mathcal{D} > 1$)?


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Is there a link of the paper you are talking about? – hwlau Feb 2 '13 at 5:35
@hwlau : – nervxxx Feb 2 '13 at 7:02
up vote 13 down vote accepted

This is a heavy question, that contains many topics in it that are worthy of their own questions, so I'm not going to give a complete answer. I am relying mainly on this excellent review paper by Nayak, Simon, Stern, Freedman and Das Sarma. The first part can be skipped by anyone already familiar with anyons.

Abelian and non-Abelian anyons

Anyons are emergent quasiparticles in two dimensional systems that have exchange statistics which are neither fermionic nor bosonic. A system that contains anyonic quasiparticles has a ground state that is separated by a gap from the rest of the spectrum. We can move the quasiparticles around adiabatically, and as long as the energy we put in the system is lower than the gap we won't excite it and it will remain in the ground state. This is partly why we say the system is topologically protected by the gap.

The simpler case is when the system contains Abelian anyons, in which case the ground state is non-degenerate (i.e. one dimensional). When two quasiparticles are adiabatically exchanged we know the system cannot leave the ground state, so the only thing that can happen is that the ground state wavefunction is multiplied by a phase $e^{i \theta}$. If these were just fermions or bosons than we would have $\theta=\pi$ or $\theta=0$ respectively, but for anyons $\theta$ can have other values.

The more interesting case is non-Abelian anyons where the ground state is degenerate (so it is in fact a ground space). In this case the exchange of quasiparticles can have a more complicated effect on the ground space than just a phase, most generally such an exchange applies a unitary matrix $U$ on the ground space (the name 'non-Abelian' comes from the fact that these matrices do not in general commute with each other).

The quantum dimension

So we know that the ground space of a system with non-Abelian anyons is degenerate, but what can we say about its dimension? We expect that the more quasiparticles we have in the system, the larger the dimension will be. Indeed it turns out that for $M$ quasiparticles, the dimension of the ground space for large $M$ is roughly $\sim d_a^{M-2}$ where $d_a$ is a number that depends on $a$ - the type of the quasiparticles in the system. This scaling law is reminiscent of the scaling of the dimension of a tensor product of multiple Hilbert spaces of dimension $d_a$, and for this reason $d_a$ is called the quantum dimension of a quasiparticle of type $a$. You can think of it as the asymptotic degeneracy per particle. For Abelian anyons we have a one-dimensional ground space no matter how many quasiparticles are in the system, so for them $d_a=1$.

Although we used the analogy to a tensor product of Hilbert spaces, note that in that case the dimension of each Hilbert space is an integer, while the quantum dimension is in general not an integer. This is an important property of non-Abelian anyons that differentiates them from just a set of particles with local Hilbert spaces - the ground space of non-Abelian anyons is highly nonlocal.

More details on anyons and the quantum dimension can be found in the review paper cited above. The quantum dimension can be generalized to other systems with topological properties, maintaining the same intuitive meaning of asymptotic degeneracy per particle. It is in general very hard to calculate the quantum dimension, and there is only a handful of papers that do (most of them cited in the paper by Kitaev and Preskill that inspired this question).

Relation to entanglement

I can also try and give a handwaving argument for why the quantum dimension would be related to entanglement. First of all, the fact that the entanglement entropy of a bounded region depends only on the length of the boundary $L$ and not on the area of the region is very clearly explained in this paper by Srednicki, which is also cited by Kitaev and Preskill. Basically it says that the entanglement entropy can be calculated by tracing out the bounded region, or by tracing out everything outside the bounded region, and the two approaches will yield the same result. This means the entanglement has to depend only on features that both regions have in common, and this rules out the area of the regions and leaves only the boundary between them.

Now for a system with no topological order the entanglement would go to zero when the size of bounded region goes to zero. However for a topological system there is intrinsic entanglement in the ground space which yields the constant term $-\gamma$ in the entanglement. The maximal entanglement entropy a system with dimension $D$ has with its environment is $\log D$, so in an analogous manner the topological entanglement is $\gamma=\log D$ where $D$ is the quantum dimension. Again this last argument relies heavily on handwaving so if anyone can improve it please do.

I hope this answers at least the main concerns in the question, and I welcome any criticism.

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@ Joe: Thank you for taking the time to research on and to answer these questions. Your answer has given me a good overview of what the quantum dimension and its relation to the e.e. is. Also the links you have provided look very helpful, and I will try to read them and understand them. If I have any further questions I'll ask you here (or in a separate qn). – nervxxx Feb 8 '13 at 18:02
@nervxxx - I'm happy to help and I welcome any questions – Joe Feb 9 '13 at 20:27
@Joe: your statement that abelian anyons exist for a non-degenerate ground state, does not seem to be true to me. Take Toric code for example, which has abelian anyons but has degenerate set of ground states. – cleanplay Sep 21 '15 at 0:58
@cleanplay: the fact that there exists a system with abelian anyons in a degenerate ground state does not contradict the statement that anyons in non-degenerate ground states are abelian. – Joe Sep 21 '15 at 5:11
@cleanplay: In a non-degenerate ground state the operators applied to the wavefunction as a result of particle exchange are multiplcations by phases. These commute with each other, therefore in a non-degenerate ground state anyons are necessarily abelian. In a degenerate ground state, the operators are Unitary matrices, which in general do not commute, therefore the anyons can be non-abelian, but are not necessarily non-abelian. Since some sets of Unitary matrices do commute, there can be a special case of anyons in a degenerate ground state which are abelian. – Joe Sep 21 '15 at 5:12

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