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I have seen many times the $BF$ theory has non-trivial ground state degeneracy (typically on torus), but I can not see how the conclusion come out. Recently I found a paper by Hansson, Oganesyan and Sondhi, Superconductors are topologically ordered in which the superconductor is described by a Maxwell-$BF$ theory. They have a section of the GSD in a $BF$ theory in $2+1$ $d$. But actually I still have questions to understand it.

The $BF$ theory in $2+1$ $d$ is given by the action $$ S = \frac{1}{\pi} \int d^3 x \epsilon^{\mu \nu \sigma} b_{\mu} \partial_{\nu} a_{\sigma}, \qquad (1) $$ where $a_{\mu}$ and $b_{\mu}$ are $U(1)$ gauge fields. $\mu,\nu,\sigma = 0,x,y$.

Working on $2$-torous, as in the section IV.A in Hansson's paper, the $BF$ theory can be written in the form $$ S = \frac{1}{\pi}\int d^3x[\epsilon^{ij} \dot{a}_i b_j+ a_0 \epsilon^{ij} \partial_i b_j + b_0 \epsilon^{ij} \partial_i a_j], $$ where $\dot{a} = \partial_0 a$ and $i,j = x,y$. They interpret $a_0$ and $b_0$ are multipliers for constraints $\epsilon^{ij} \partial_i b_j = 0$ and $\epsilon^{ij} \partial_i a_j = 0$. Upon inserting $a_i = \partial_i \Lambda_a + \bar{a}_i/L$ and $b_i = \partial_i \Lambda_b + \bar{b}_i/L$, where $\Lambda_{a/b}$ are periodic functions on the torus, $\bar{a_i}$ and $\bar{b_i}$ are spatially constant, $L$ denotes the size of the system, the above $BF$ theory reduces to $$ S = \frac{1}{\pi}\int d^3 x \epsilon^{ij} \dot{\bar{a}}_i \bar{b}_j. \qquad (2) $$

Then they say from the Eq.(2) one can obtain the commutation relation ( Eq. (38) in their paper) $$ [\bar{a}_x, \frac{1}{\pi}\bar{b}_y] = i, \quad [\bar{a}_y,-\frac{1}{\pi}\bar{b}_x] = i. \qquad (3) $$

Moreover, from the commutation relations Eq. (3), one can have ( Eq. (39) in their paper)

$$ A_x B_y + B_y A_x = 0, \quad A_y B_x + B_x A_y = 0, \qquad (4) $$ where $A_i = e^{i\bar{a}_i}$ and $B_i = e^{i\bar{b}_i}$. They claim that relations Eq. (4) indicates a $2\times2 = 4$-fold GSD and "$B_i$ can be interpreted either as measuring the $b$-flux or inserting an $a$-flux."

There are several points that I don't understand.

  1. How can I get communication relations Eq. (3) from the action Eq. (2)?
  2. Why relations Eq. (4) indicate a $4$-fold GSD?
  3. How should I understand the statement "$B_i$ can be interpreted either as measuring the $b$-flux or inserting an $a$-flux."?

I would be very appreciate if anyone can give me some hints or suggest me some relevant references.

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A small comment first: usually people call this theory as a Chern-Simons theory in (2+1)d, while the BF theory usually refers to a similar theory in (3+1)d. But anyways, this naming is not important. The U(1) Chern-Simons theory in (2+1)d is always formulated in the following general form $$S=\int\frac{\mathrm{i}}{4\pi}K^{IJ} a^I\wedge \mathrm{d}a^J.$$ The action in your Eq. (1) corresponds to the following $K$ matrix $$K=\left[\begin{matrix}0&2\\2&0\end{matrix}\right],$$ which is a very famous $K$ matrix that corresponds to the $\mathbb{Z}_2$ toric-code topological order (or $\mathbb{Z}_2$ gauge theory). The low-energy effective theory for a fully gapped superconductor is a $\mathbb{Z}_2$ gauge theory, simply because the condensation of charge-2 Cooper pair has Higgsed the U(1) gauge structure to $\mathbb{Z}_2$. This is equivalent (and probably better) to say that the superconductor exhibits $\mathbb{Z}_2$ topological order.

Regarding your technical questions, I would recommend Wen and Zee's original paper (cond-mat/9711223) on the topological ground state degeneracy of the Chern-Simons theory, or section 8.2.1 of Wen's book. Your last question is related to the Kitaev toric-code model, which I would suggest Kitaev's original paper (cond-mat/0506438).

Let me try to briefly answer you questions.

  1. From Eq.(2) we known the Lagrangian $L=\frac{1}{\pi}(\dot{\bar{a}}_x\bar{b}_y-\dot{\bar{a}}_y\bar{b}_x)$, according to classical mechanics, the conjugate momenta of $\bar{a}_x$ and $\bar{a}_y$ are $$p_{\bar{a}_x}=\frac{\partial L}{\partial(\dot{\bar{a}}_x)}=\frac{1}{\pi}\bar{b}_y,\quad p_{\bar{a}_y}=\frac{\partial L}{\partial(\dot{\bar{a}}_y)}=-\frac{1}{\pi}\bar{b}_x.$$ By canonical quantization ($[q,p]=\mathrm{i}$ in quantum mechanics), Eq.(3) simply follows from $$[\bar{a}_x,p_{\bar{a}_x}]=[\bar{a}_x,\frac{1}{\pi}\bar{b}_y]=\mathrm{i},\quad [\bar{a}_y,p_{\bar{a}_y}]=[\bar{a}_y,-\frac{1}{\pi}\bar{b}_x]=\mathrm{i}.$$

  2. To calculate the ground state degeneracy along this line of though, one need to known that the gauge fields $a$ and $b$ are both compact due to the fact that their gauge charges are quantized (see section 6.3 of Wen's book), which means that besides the local gauge transformation, the so-called large gauge transformation is also allowed. On the torus, the large gauge transformation send $\bar{a}_i\to\bar{a}_i+2\pi$ and $\bar{b}_i\to\bar{b}_i+2\pi$. Gauge configurations that are related by the gauge transformation are just different labels of the same physical quantum state, so the large gauge transformation actually imposes the boundary condition on $\bar{a}$ and $\bar{b}$. For example, $|\bar{a}_x\rangle=|\bar{a}_x+2\pi\rangle$ are the same state. So the quantum mechanical wave function is subject to the periodic boundary condition like $\psi(\bar{a}_x)=\psi(\bar{a}_x+2\pi)$, meaning that the momentum $p_{\bar{a}_x}=\frac{1}{\pi}\bar{b}_y$ must be quantized to an integer (recall the momentum quantization formula $\frac{2\pi n}{L}$ with $L=2\pi$), i.e. $\bar{b}_y=n\pi$ (with $n\in\mathbb{Z}$). However by the large gauge transformation $|\bar{b}_y\rangle=|\bar{b}_y+2\pi\rangle$ are also the same state, so $n=0,1$ can only take two values, which corresponds to two eigen states of $\bar{b}_y$, spanning a 2-dim Hilbert space. Repeat the same argument for the other conjugate pair $\bar{a}_y$ and $\bar{b}_x$, one can find another 2-dim Hilbert space. Eventually the ground states Hilbert space is simply the direct product of both 2-dim Hilbert spaces, which contains 4 states in total, hence the 4-fold GSD.

  3. You can understand the statement in the same way as you understand the following statement in quantum mechanics: the momentum operator $p$ measures the momentum of a particle, and also generates the coordinate translation. Every quantum mechanical operator has two effects: measuring and operating. If it measures the momentum, it must also operate (or change) the coordinate. Now the relation between $a$-flux and $b$-flux is just like the relation between coordinate and momentum, so any operator that measures $b$-flux must necessarily change the $a$-flux (by inserting the flux of cause), and obviously $B$ is such an operator. In the toric-code model, $A$ and $B$ are also known as the loop operator along the homology basis, which has a clearer geometrical and physical meaning. For more about the loop algebra, you may look at the paper (1208.4834) by Barkeshli, Jian and Qi, or the paper (1208.4109) by You, Jian and Wen.

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