# effective field theory of the projective semion model

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: $$\mathcal{L} = \frac{2}{4\pi}\epsilon^{\mu\nu\lambda}a_{\mu}\partial_{\nu}a_{\lambda}$$

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

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

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 $\mathbb{Z}_2\times\mathbb{Z}_2$?

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.
• 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. Mar 26, 2015 at 3:07
• 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. Mar 26, 2015 at 3:10
• 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 $a\rightarrow a+df$. Mar 26, 2015 at 3:26
• Under $a \rightarrow a+df$, $\delta L$ is just a bunch of total derivatives, hence $S$ is invariant under the gauge transformation of $a$. Mar 26, 2015 at 3:30
• 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 $dA_1=dA_2=0$. Mar 26, 2015 at 3:34