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

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?

  • 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. – Zitao Wang Mar 26 '15 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. – Zitao Wang Mar 26 '15 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 $a\rightarrow a+df$. – Meng Cheng Mar 26 '15 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 $a$. – Zitao Wang Mar 26 '15 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 $dA_1=dA_2=0$. – Meng Cheng Mar 26 '15 at 3:34

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