# effective field theory of the projective semion model

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The "projective semion" model was considered in http://arxiv.org/abs/1403.6491 (page 2). It is a symmetry enriched topological (SET) phase. There is one non-trivial anyon, a semion $s$ which induces a phase factor of $\pi$ when going around another semion.The chiral topological order is the same as the $\nu = 1/2$ bosonic fractional quantum Hall state, whose effective field theory is the $K = 2$ Chern-Simons theory: \begin{equation} \mathcal{L} = \frac{2}{4\pi}\epsilon^{\mu\nu\lambda}a_{\mu}\partial_{\nu}a_{\lambda} \end{equation}

The symmetry group for the theory is $G = \mathbb{Z}_2 \times \mathbb{Z}_2$. We label the three non-trivial group elements as $g_x, g_y, g_z$. The symmetry can act on the semion in the following ways:

1. Each semion carries half charge for all three $\mathbb{Z}_2$ transformations. Moreover the three $\mathbb{Z}_2$ transformations anticommute with each other and can be represented as $g_x = i\sigma_x, g_y = i\sigma_y, g_z = i\sigma_z$.

2. The semion carries integral charge under two of the three $\mathbb{Z}_2$ transformations, and half charge under the the other $\mathbb{Z}_2$ transformation. There are three variants of this, and the symmetry group can be represented as $g_x = \sigma_x, g_y = \sigma_y, g_z = i\sigma_z$, or $g_x = \sigma_x, g_y = i\sigma_y, g_z = \sigma_z$, or $g_x = i\sigma_x, g_y = \sigma_y, g_z = \sigma_z$.

Symmetry fractionalization in case 1 is anomaly free but is anomalous in case 2, as shown in http://arxiv.org/abs/1403.6491.

I want to write down an effective field theory description to describe symmetry fractionlization pattern in cases 1 and 2 on the semion $a$, and can explicitly see that the field theory I write down for case 1 is anomaly free whereas that for case 2 has an anomaly.

One possible way is to gauge the symmetry $\mathbb{Z}_2 \times \mathbb{Z}_2$, and couple the gauge fields to the semion $a$. The different coupling terms reflect the different ways that the symmetry is represented on the semion. I think this is essentially what Eq.(5) on page 21 of http://arxiv.org/abs/1404.3230 is trying to describe. The action they wrote down is

\begin{equation} \mathcal{L} = \frac{2}{4\pi}\epsilon^{\mu\nu\lambda}a_{\mu}\partial_{\nu}a_{\lambda} + \frac{p_1}{2\pi}\epsilon^{\mu\nu\lambda}a_{\mu}\partial_{\nu}A_{1\lambda} + \frac{p_2}{2\pi}\epsilon^{\mu\nu\lambda}a_{\mu}\partial_{\nu}A_{2\lambda} + \frac{p_3}{\pi^2}\epsilon^{\mu\nu\lambda}a_{\mu}A_{1\nu}A_{2\lambda} \end{equation}

I can understand the second and third terms in this action, which says (with $p_1=p_2=1$) that the semion $a$ carries half symmetry charge under the two generators (say $g_x$ and $g_y$) of $\mathbb{Z}_2\times \mathbb{Z}_2$.

However, I am having trouble understanding the last term in the action, presumably, it means that the semion carries half charge under all three elements $g_x,g_y,g_z$ in $\mathbb{Z}_2\times\mathbb{Z}_2$. If this is correct then setting $p_1=p_2=0, p_3=1$ gives us an effective description of case 1. The theory is anomaly free; whereas setting $p_1=p_2=p_3=1$ gives us an effective description of case 2 (semion $a$ carries half $g_x,g_y,g_z$ charge from the last term, and an additional half $g_x,g_y$ charge from the second and third term), and the theory is anomalous. This is consistent with the claim on page 24 of http://arxiv.org/abs/1404.3230.

Does any people have an idea why the last term in $\mathcal{L}$ says that the semion carries half charge under all three elements $g_x,g_y,g_z$ in $\mathbb{Z}_2\times\mathbb{Z}_2$?

This post imported from StackExchange Physics at 2015-04-27 21:18 (UTC), posted by SE-user Zitao Wang

edited Apr 27, 2015

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It might be useful to consider the physical meaning of the term $aA_1A_2$ in a gauge theory. Compactify the theory on a "thin" torus, say the length of the $y$ direction $l_y$ is much smaller than $l_x$. The two ground states are distinguished by the value of the Wilson loop along $y$. Heuristically, we can just substitute $a=0,\pi$ (I'm sloppy about the indices...), and in the semion sector we get a term $A_1\wedge A_2$ in the "dimensionally reduced" $1+1$ theory. As described in http://arxiv.org/abs/1401.0740, this is the continuum version of the $1+1$ Dijkgraaf-Witten theory of $\mathbb{Z}_2\times\mathbb{Z}_2$ gauge field, and describe the 1D SPT protected by this symmetry. This implies that a semion is the end of a 1D $\mathbb{Z}_2\times\mathbb{Z}_2$ SPT, which carries spin-1/2 (or the semions Wilson line is "decorated" by a Haldane chain). However, since 1D SPT are classified by $H^2(G, U(1))$, it is ambiguous about the particular class in $H^2(G, Z_2)$, which is really what your question is about.

So this argument is certainly not satisfactory and does not really address your question directly. Maybe going to edge theory and figure out the symmetry transformation on the edge modes could help?

This post imported from StackExchange Physics at 2015-04-27 21:18 (UTC), posted by SE-user Meng Cheng
answered Mar 26, 2015 by (550 points)
Thanks! Yes, I think dimension reduction is a good way to see the physics. But I'm not sure if there is a more direct way to interpret the last term. In particular, what confuses me is that in page 22 of the paper arxiv.org/abs/1404.3230, they stipulate that if we consider the $\mathbb{Z}_2 \times \mathbb{Z}_2$ as arising from a subgroup of $U(1) \times U(1)$, then the semion $a$ would transform under the gauge $U(1) \times U(1)$ in a rather bizarre way $a \rightarrow a-q_1f_1\frac{d\phi_2}{2\pi}$. This seems kind of artificial to me and I don't have a good intuition for that.

This post imported from StackExchange Physics at 2015-04-27 21:18 (UTC), posted by SE-user Zitao Wang
Also I'm not sure about what you mean by going to the edge theory. This theory lives on the boundary of a $3+1$d SPT, whose effective action is a DW action $S_4$ on page 23 of arxiv.org/abs/1404.3230. And the boundary of a boundary should vanish.

This post imported from StackExchange Physics at 2015-04-27 21:18 (UTC), posted by SE-user Zitao Wang
If $p_1=p_2=0$ the theory is anomaly free and can be realized in 2D (in fact a chiral spin liquid is a perfect example, with the $Z_2\times Z_2$ symmetry being the $\pi$ rotations around the $x,y,z$ axes). So there is no problem talking about the edge theory. Only the anomalous ones need a 3D SPT bulk to regularize. Regarding the issue of gauge invariance, it seems that $a\rightarrow a - q_1f_1A_2$ is just postulated to cancel the variation of the $aA_1A_2$. Actually, I'm not sure how the action is invariant under the gauge transformation $a\rightarrow a+df$.

This post imported from StackExchange Physics at 2015-04-27 21:18 (UTC), posted by SE-user Meng Cheng
Under $a \rightarrow a+df$, $\delta L$ is just a bunch of total derivatives, hence $S$ is invariant under the gauge transformation of $a$.

This post imported from StackExchange Physics at 2015-04-27 21:18 (UTC), posted by SE-user Zitao Wang
So I get $df A_1 A_2$ from the last term. How is this a total derivative? It can match $d(fA_1A_2)$, but only under the flat connection assumption $dA_1=dA_2=0$.

This post imported from StackExchange Physics at 2015-04-27 21:18 (UTC), posted by SE-user Meng Cheng
Yeah if the effective action when $p_1=p_2=0$ describes a chiral spin liquid then this effectively answers my confusion. In a CSL, semion carries half charge for each of the three $\mathbb{Z}_2$ transformations. This means that my naive guess of the meaning of the last term is correct.

This post imported from StackExchange Physics at 2015-04-27 21:18 (UTC), posted by SE-user Zitao Wang
It's not just $A_1$ $A_2$ its self, it's $nA_1$ and $nA_2$. And $ndA_1 = ndA_2 = 0$. So $A$ is not flat, but an $n-$torsion.

This post imported from StackExchange Physics at 2015-04-27 21:18 (UTC), posted by SE-user Zitao Wang
Written in their original notation page 21. The last term is $\frac{p_3}{(2\pi)^2}ad\phi_1d\phi_2$, where $\phi$'s are the Higgs fields to Higgs $U(1)$ down to $\mathbb{Z}_n$ ($n=2$ in our case) by requiring that $d\phi_i = nA_i$. And written in the form $\frac{p_3}{(2\pi)^2}ad\phi_1d\phi_2$, it's very easy to see that under $a \rightarrow a+df$, this term contributes a total derivative.

This post imported from StackExchange Physics at 2015-04-27 21:18 (UTC), posted by SE-user Zitao Wang
I forgot the $n$ in the expression. This makes a lot sense.

This post imported from StackExchange Physics at 2015-04-27 21:18 (UTC), posted by SE-user Meng Cheng
I think the transformation law that they postulate $a \rightarrow a -q_1f_1\frac{d\phi_2}{2\pi}$ under $U(1) \times U(1)$ is closely related to how $\mathbb{Z}_2 \times \mathbb{Z}_2$ is represented on the semions. If we can make sense why $a$ should transform like this, we know how $\mathbb{Z}_2 \times \mathbb{Z}_2$ is represented on the semion.

This post imported from StackExchange Physics at 2015-04-27 21:18 (UTC), posted by SE-user Zitao Wang
Also does the action with $p_1=p_2=0$ describe a chiral spin liquid? We expect this because it is anomaly free, and the effective action with $p_1=p_2=0$ is anomaly free. The transformation law of $a$ under $U(1) \times U(1)$ is sort of by construction making the third term by itself gauge invariant (hence anomaly free).

This post imported from StackExchange Physics at 2015-04-27 21:18 (UTC), posted by SE-user Zitao Wang
The action of a chiral spin liquid can be written down using $CP^1$ representation: $\frac{2}{4\pi}ada+v|(i\partial-a-\mathbf{A}\cdot \sigma)z|^2$ where $z=(z_\uparrow, z_\downarrow)^T$, and $\mathbf{A}$ is the $SO(3)$ gauge field, which can be further broken down to $Z_2\times Z_2$. Not sure how this can be related to the $p_3$ term...

This post imported from StackExchange Physics at 2015-04-27 21:18 (UTC), posted by SE-user Meng Cheng
Thanks! Perhaps one can introduce one more ghost field to Higgs $SO(3)$ down to $\mathbb{Z}_2 \times \mathbb{Z}_2$. I'll try to work on it and see if it makes sense.

This post imported from StackExchange Physics at 2015-04-27 21:18 (UTC), posted by SE-user Zitao Wang
Hi Meng, I was a bit confused about the $\mathbb{Z}_2$ coefficient appearing in the group cohomology. I thought that the projective representations are classified by $H^2(G,U(1))$.

This post imported from StackExchange Physics at 2015-04-27 21:18 (UTC), posted by SE-user Zitao Wang
For symmetry fractionalization of anyons, one needs to use the group of Abelian anyons as coefficients. In fact, in the projective semion example, case 1 and case 2, if regarded as $U(1)$ 2-cocycles, are cohomologically equivalent. And we know $H^2(Z_2, U(1))=0$, but semions can still carry half charge of $Z_2$ because $H^2(Z_2, Z_2)=Z_2$. The reason to use Abelian anyons as coefficients is the consistency of the fractionalization with fusion rules of anyons, as explained in Sec. IV of arxiv.org/pdf/1410.4540.pdf.

This post imported from StackExchange Physics at 2015-04-27 21:18 (UTC), posted by SE-user Meng Cheng
I see. Thanks you.

This post imported from StackExchange Physics at 2015-04-27 21:18 (UTC), posted by SE-user Zitao Wang
Is there a way to tell the symmetry fractionalization on the semion in $2+1$d (a class in $H^2(\mathbb{Z}_2\times\mathbb{Z}_2,\mathbb{Z}_2)$)from the symmetry action on the semion on edge of the dimensionally reduced $1+1$d SPT (in this case it's pretty trivial, because the edge spin-1/2 of the semion carries half charge under $\mathbb{Z}_2$)? I guess mathematically, my question is equivalent to ask if there is a homomorphism from $\mathbb{Z}_2=H^2(\mathbb{Z}_2\times\mathbb{Z}_2,U(1))\rightarrow H^2(\mathbb{Z}_2\times\mathbb{Z}_2,\mathbb{Z}_2)=\mathbb{Z}_2^3$ (Hopefully an injective one).

This post imported from StackExchange Physics at 2015-04-27 21:18 (UTC), posted by SE-user Zitao Wang
I don't think so. Among the 8 classes in $H^2(Z_2\times Z_2, Z_2)$, four of them map to the nontrivial class in $H^2(Z_2\times Z_2, U(1))$. I don't see a way to invert the mapping.

This post imported from StackExchange Physics at 2015-04-27 21:18 (UTC), posted by SE-user Meng Cheng
Hi Meng, I think I understand the physics of $aA_1A_2$ now. First this action corresponds to type III cocycles $\omega(A,B,C) = exp(i\pi a_1b_2c_3)$in the DW model with $\mathbb{Z}_2^3$ symmetry (page 10, Table I in Juven's paper exp(i\pia_1b_2c_3)). Then the statistics of different cocycles in the $2+1$d DW model is considered in Chenjie's paper (page 9 of arxiv.org/abs/1412.1781). The statistics described by $aA_1A_2$ is non-Abelian. It says that after a "Borromean" shaped braiding between $a,A_1,A_2$ fluxes, we get an phase of $\Theta_{123}=\pi$.

This post imported from StackExchange Physics at 2015-04-27 21:18 (UTC), posted by SE-user Zitao Wang
I think in this case, only the $adA_1$ and $adA_2$ term contributes to the mutual statistics $(a,A_1)$, $(a,A_2)$. In our case, the 2nd and 3rd terms say that $a$ carries half $A_1$ and $A_2$ charge. However, the fourth term $aA_1A_2$ only contributes to the non-Abelian "Borromean"-braided statistics, and does not contribute to the mutual statistics $(a,A_1)$, $(a,A_2)$. So if we only have the 4h term in the action (i.e. $p_1=p_2=0, p_3=1$), we should not see any non-trivial mutual statistics $(a,A_1)$, $(a,A_2)$. In particular, the notion that $a$ carries half $A_1$, $A_2$ charge is wrong.

This post imported from StackExchange Physics at 2015-04-27 21:18 (UTC), posted by SE-user Zitao Wang
Yeah, I also noticed the similarity to the type III cocycle, although it was a SPT phase there and now we have a SET. Whilte I'm not entirely sure you can directly apply the argument from Chenjie's paper, the result makes sense. But how does that imply that semion must carry half charge of all three generators of $\mathbb{Z}_2^2$?

This post imported from StackExchange Physics at 2015-04-27 21:18 (UTC), posted by SE-user Meng Cheng
Yeah. I think in this case, it is incorrect to say that the semion carry half charge of all three generators.

This post imported from StackExchange Physics at 2015-04-27 21:19 (UTC), posted by SE-user Zitao Wang
Wait...I think we are all convinced that the non-anomalous theory must have the semion carrying half charge of all three generators, from physical constructions of chiral spin liquid and the $H^4$ construction in Chen et. al.

This post imported from StackExchange Physics at 2015-04-27 21:19 (UTC), posted by SE-user Meng Cheng
I think this simply says that this effective action does not describe the projective semion model?

This post imported from StackExchange Physics at 2015-04-27 21:19 (UTC), posted by SE-user Zitao Wang
Does it make sense to talk about mutual statistics $(a,A_1)$, $(a,A_2)$ in $aA_1A_2$? I think you'll only get trivial mutual statistics.

This post imported from StackExchange Physics at 2015-04-27 21:19 (UTC), posted by SE-user Zitao Wang
I don't really know...it is not clear that $aA_1A_2$ only contributes to the Borromean statistics. It definitely does this, from several arguments (dimension reduction, similarity with type III, etc.), but I don't think we can conclude that it does not affect the half-charge of semions. But I don't fully understand the field theory arguments in Kapustin's paper either...

This post imported from StackExchange Physics at 2015-04-27 21:19 (UTC), posted by SE-user Meng Cheng
We should continue this discussion somewhere else. Do you have a gtalk/skype?

This post imported from StackExchange Physics at 2015-04-27 21:19 (UTC), posted by SE-user Meng Cheng
hm...Yeah I'm not sure how to compute mutual statistics for this term. I'll have to think. I think the reason why Kapustin add this term is that in general the action would contain couplings between little a and the Higgs fields $\phi_1$, $\phi_2$, which Higges $U(1)\times U(1)$ down to $Z_2\times Z_2$.

This post imported from StackExchange Physics at 2015-04-27 21:19 (UTC), posted by SE-user Zitao Wang
yeah I did. my gtalk is zwangab91@gmail.com.

This post imported from StackExchange Physics at 2015-04-27 21:19 (UTC), posted by SE-user Zitao Wang
This is also my Skype account. We could probably arrange some time that works for both of us. Thanks!

This post imported from StackExchange Physics at 2015-04-27 21:19 (UTC), posted by SE-user Zitao Wang

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