effective field theory of the projective semion model

Question

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

Answer 1

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

Comments

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

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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

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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

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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

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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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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

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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