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Topological entanglement entropy (, is usually calculated for surfaces with boundary. How would it look like for compact surfaces and when these are punctured?

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up vote 7 down vote accepted

In the absents of any response, let me try to give a quick answer.

I am a little bit confused about why you say the topological entanglement entropy (TEE) is usually calculated on surfaces with boundaries. You can, and I think this is whats usually done, calculate it on compact manifolds. A boundary will always be present since you need to do a bi-partition of the manifold to calculate the entanglement entropy (EE). If you actually did the calculation a manifold with boundary, the gapless degrees of freedom on the boundary might cause some trouble extracting the TEE. Let me try to briefly describe what happens in different situations.

Assume you put a system with a gap above the ground state, on a (either compact, or non-compact with no boundary) two-dimensional manifold $\mathcal M$, and then cut $\mathcal M$ into two contractible submanifolds $\mathcal A$ and $\mathcal B$. Given the reduced density matrix of the ground-state in the subsystem $\mathcal A$, $\rho_\mathcal A$ (given by tracing out the information in $\mathcal B$), the entanglement entropy is given by $ S_\mathcal A = -\text{tr}\left(\rho_\mathcal A\log\rho_\mathcal A\right)$. As shown by Preskill-Kitaev and Levin-Wen (as you cite), the EE has the following form

$$ S_\mathcal A = \alpha L_\mathcal A - \gamma + \dots, $$ where $\alpha$ is a non-universal number, $L_\mathcal A$ is the boundary area $\partial\mathcal A$, and '$\dots$' are terms which vanish in the limit $L\rightarrow\infty$. The constants piece, $\gamma$, is universal and what we call the TEE. Its determined purely by topological data $$ \gamma = \log\mathcal D =\log\sqrt{\sum_id^2_i}, $$ where $d_i$ is the quantum dimension of the $i$'th topological quasiparticle and $\mathcal D$ is the total quantum dimension. If this calculation is done in the purely topological field theory limit in the infrared, only the topological piece will survive. If one instead calculates the Renyi entropies, $S^n_\mathcal A = \frac 1{1-n}\log\left({\text{tr}\rho^n_\mathcal A}\right) = \alpha_n L_\mathcal A - \gamma +\dots$, the topological piece will be independent of $n$ and no new information gained.

  1. This is all the standard story, as mentioned in the papers you cite. One could ask what happens if the bipartition is not contractible? For example, say $\mathcal M=T^2$ is the 2-torus and you make a cut such that $\mathcal A$ is not simply-connected (see figure 1 of ref. 1). In this case it turns out that the topological piece, $\gamma_n'$, in the Renyi entropy $$ S^n_\mathcal A = \alpha_nL_\mathcal A - \gamma_n' + \dots,$$ will besides $\gamma = \log\mathcal D$, depend on $n$ and the coefficients $c_j$ of the ground state in a special basis $|\Phi\rangle = \sum_jc_j|\Theta_j\rangle$. For the detailed formula, see equation (2) in ref 1 and equation (2.38) of ref. 2. The basis states $|\Theta_i\rangle$ are called minimum entropy states (MESs), see 1 for detailed definition.
  2. Now what happens if $\mathcal M$ is a punctured manifold? By giving appropriate boundary conditions, these punctures essentially correspond to quasi-particles. Thus in this case, we are calculating the entanglement entropy in the presence of excitations. Here it turns out that, depending on the particular partition, the result will depend on various components of the topological $\mathcal S$-matrix. This means that in this way, we can extract much more topological information of the underlying TQFT than just the total quantum dimension. For details, see section 3 of 2.

EDIT: In the comments, Hamurabi ask the interesting question of what happens in higher dimensions. Let me briefly mention the proposal of ref 4:

Under certain general assumptions, one can write the (Von Neumann) EE as the sum of two pieces

$$S_\mathcal A = S_{\mathcal A,local} + S_{\mathcal A,topological},$$

where the first term depends on local information of the system while the latter encode the global, topological piece of entanglement. They argue, that the local piece in dimension $D$ has the following expansion

$$ S_{\mathcal A,local} = \alpha_1L_\mathcal A ^{D-1} + \alpha_3L_\mathcal A^{D-3}+\alpha_5L_\mathcal A^{D-5} + \dots,$$ where all $\alpha_i$'s are non-universal. Note that in $D$ even, $S_{\mathcal A,local}$ does not have any constant piece and any constant piece of $S_\mathcal A$ must therefore be topological (as in $D=2$). For $D$ odd, however, there can be non-topological constant piece and thus make it a little harder to extract $S_{\mathcal A,topological}$. They propose (and check in several examples) the following general form of the TEE

$$S_{\mathcal A,topological} = \begin{cases}-\gamma_0 b_0 - \gamma_1 b_1 - \dots -\gamma_{\frac D2-1}b_{\frac D2-1}, \qquad &\text{if}\; D\; \text{is even},\\ -\gamma_0 b_0 - \gamma_1 b_1 + \dots -\gamma_{\frac {D-3}2}b_{\frac {D-3}2}, \qquad &\text{if}\; D\; \text{is odd}, \end{cases}$$ where $b_i$ is the $i$'th Betti number of the manifold $\partial A$. For example the general expression in $D=2$ is $S_\mathcal A = \alpha_1 L_\mathcal A - b_0\gamma_0$, where the zeroth Betti number $b_0$ just counts the number of connected components of $\partial A$. Notice that for $D=2,3$ there is only one type of TEE, while there are several in higher dimensions!


[ 1 ] Zhang et al, Quasi-particle Statistics and Braiding from Ground State Entanglement Phys. Rev. B 85, 235151 (2012), arXiv:1111.2342

[ 2 ] Dong et al, Topological Entanglement Entropy in Chern-Simons Theories and Quantum Hall Fluids JHEP05(2008)016, arXiv:0802.3231

[ 3 ] Hikami, Skein Theory and Topological Quantum Registers: Braiding Matrices and Topological Entanglement Entropy of Non-Abelian Quantum Hall States arXiv:0709.2409

[ 4 ] Grover et al, Entanglement Entropy of Gapped Phases and Topological Order in Three dimensions Phys. Rev. B 84, 195120 (2011), arXiv:1108.4038

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Thank you for the comprehensive answer. Firstly, indeed you want a bi-partition. For a torus or a sphere you would want to have it embedded in some 3-space and the boundary is the surface $A$ of these two manifolds. Preskill, Kitaev base their discussion on a disc with boundary length $L$. Would this lead to something like $S=\alpha A - \gamma$? In 1. what do you mean by $n$? Your second reference cites on page 4: They claim that they get the same results and you might want to add it to your list of references. – Hamurabi Aug 19 '13 at 16:12
@Hamurabi I am not sure I understand why you want to embed the torus and sphere in some 3-space? Maybe I am misunderstanding where your question a little? When Preskill-Kitaev (before eq. 1) talk about a disc, (I think) they mean they have a theory on $\mathcal M=\mathbb R^2$ and they choose a bi-partition such that $\mathcal A$ is a disc with boundary length $L$ and $\mathcal B$ is the complement (which is traced out). – Heidar Aug 20 '13 at 6:51
What I mean by $n$, is that $S^n_\mathcal A$ is the $n$'th Renyi entropy. In the case of $n=1$, we get the usual Von Neumann entropy. I used the more general Renyi entropy, to illustrate that for non-contractible partitions, the constant piece $\gamma_n'$ is dependent on $n$ which is normally not the case. I will add the reference, thanks. – Heidar Aug 20 '13 at 6:52
The question would maybe read: Is it possible to generalize the expression $S_\mathcal A = \alpha L_\mathcal A - \gamma + \dots$ to higher dimensions? – Hamurabi Aug 20 '13 at 15:36
@Hamurabi Ah I see, thats a slightly different question. My understanding is that this expression does generalize to higher dimensions (atleast to $D=3$) but there are some subtleties which I don't know much about. Such that in even dimensions, the constant piece $\gamma$ only contain topological information, while in odd dimensions (such as $D=3$) it can be non-zero even for trivial phases. You can read the details in this paper . – Heidar Aug 21 '13 at 0:30

I would like to address a possible complexity here for the manifold with boundary, such as the punctures on the manifold. The first thing to ask is whether the edge states are gapless or gapped. The situation may be different. Here let me say something about the gapped boundary case. It has been recently studied:

As discussed by Heidar and Hamurabi, Topological Entanglement Entropy(TEE) $S_{TEE}=-\log D$. But more explicitly, we can write, at least for Abelian topological order,

$$ D=e^{-S_{TEE}}= \text{quantum dimension of the system}\\ = \text{number of quasiparticle types of the system}\\ =\text{ground state degeneracy(GSD) of the system on a $T^2$ torus} $$

This tells us $D$ is related to the ground state degeneracy(GSD) of the system on the torus. There is some intuitive picture using string-net or Wilson line(line operator) of quasiparticles(anyons) to count this GSD, thus the TEE. One may ask whether the GSD of topological order may depend on the manifold with gapped boundaries?

The answer is absolute yes. Let us take a sphere with two punctures. It is found, in 1212.4863, $Z_2$ toric($Z_2$ gauge theory) code has GSD=2 or GSD=1 depending on the types of gapped boundaries, while $Z_2$ doubled semions(twist $Z_2$ gauge theory) has GSD=2 regardless the types of gapped boundaries. One can again use the string-net or Wilson line of anyons to count GSD, explained in 1212.4863.

Thus you make ask further whether this GSD property will affect the TEE for the manifold with gapped boundaries?

We would not be too surprise if there is indeed this possibility.

ps. Actually some months ago I did have some thoughts on working out Topological Entanglement Entropy for this kind of generic cases for, the manifold with gapless/gapped boundaries and punctures.

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It could indeed be interesting to investigate whether the sensitivity of GSD on the type of gapped boundary also translate into such a sensitivity in TEE. Very nice comment. – Heidar Aug 21 '13 at 18:46

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