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It is well-known that integer and fractional quantum Hall effect can be effectively described by $U(1)$ abelian Chern-Simon theory. In both cases, quantization(fractionalization) of Hall resistance is depended on level quantization of Chern-Simon theory.

$$ \frac{k}{4\pi}\int_W AdA $$ where $W$ is our three manifold. As I saw in literatures, for example at page 151, https://arxiv.org/abs/1606.06687, they look at the partition function of the theory at the finite temperature and take the spatial directions as a sphere $S^2$. So the total $2+1$ space-time is $W=S^2*S^1$ and it is easy to see that the level $k$ must be an integer for consistency of the quantum theory if we have a unit flux threading through the sphere.

$$ \frac{1}{2\pi}\int_{S^2} F=\frac{h}{e} $$

my question is that, in real situation, our sample is not a sphere and usually is a rectangular and (I think) experiments are performed at zero temperature. How Chern-Simon level have to be quantized in $W=\mathbb{R^2}*\mathbb{R}$ background.

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(this is an argument coming from path integral quantization. There are more succinct arguments coming from canonical quantization).

The quantization of the Chern-Simons level comes from a requirement of gauge invariance at the quantum level. One can show that the variation of the action is something like $$\delta S= \text{boundary term} + 2\pi i k *\text{winding number of gauge transformation}$$

When $M=\mathbb{R}^3$, there are no large gauge transformations (that is, the winding number of all transformations is always zero) so $k$ can be any real number. This is unlike the case of other spacetimes, such as the one you are describing, where the winding number can be any integer, hence we require the quantization of $k$ to have a variation that is a integer multiple of $2\pi i$, so that the partition function does not change.

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  • $\begingroup$ thanks, yes I know that, but as I explained, I think that for the real samples in experiments, the appropriate manifold is $\mathbb{R^2}*\mathbb{R}$ and the quantization is not required. but in literatures, quantization of resistance is referred to level quantization of Chern-Simon theory. It seems that I have missed something. $\endgroup$
    – Arian
    Jun 15, 2021 at 18:20
  • $\begingroup$ Usually, in practice, you require the gauge transformations to be well-defined at infinity. Introducing these boundary conditions is ultimately equivalent to working on the 3-sphere. $\endgroup$
    – Alonso Perez Lona
    Jun 15, 2021 at 18:39
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The rectangle case is easier than the sphere as one can take $$\Phi_x= \int_{\alpha} A_x dx\\\Phi_y= \int_\beta A_y dy$$ as the gauge invariant degress of freedom. Here $\alpha$ and $\beta$ are the one-cycles generating $H_1[T^2]$. Then, if I remember correctly (it's been a while) the algebra of the commutator $$ [\Phi_x,\Phi_y]= \frac{2\pi i}{k} $$ together with the compactness of the range of the $\Phi$'s requires an integer $k$.

The phase space is itself a torus and so the Hilbert space is finite dimensional and can be spanned by Theta functions of level $k$.

A recent reference is this.

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  • $\begingroup$ thanks, are there exist any cycle for rectangle? $\endgroup$
    – Arian
    Jun 15, 2021 at 17:57
  • $\begingroup$ by rectangle I meant torus. The inside of the rectangle is not very interesting topologically. The pure CS on the reactangle only has degrees of the freedom on the boundary where you have chiral bosons. Look up K-matrix and quantum Hall for the CS-> Chiral boson connection. $\endgroup$
    – mike stone
    Jun 15, 2021 at 21:59
  • $\begingroup$ my question is as follow, the real sample is something like rectangle, and quantum Hall effect is performed at zero temperature(I am not sure about it), so our manifold has not intersting topology for having integer level k. If our spatial directions form a sphere or (a compact surface) and we look at the theory at finite temperature, I have no problem with level quantization. $\endgroup$
    – Arian
    Jun 15, 2021 at 23:21
  • $\begingroup$ We know that resistance is quantize on rectangle at zero temperature, but when we look at this from Chern-Simon theory, I don't know why level $k$ should be an integer. Because rectangle is topologically trivial and at zero temperature, we don't need $S^1$ from time periodicity if we look at the thermal partition function, so our manifold is $R^2×R$ instead of $S^2×S^1$ and we have not level quantization for Chern-Simon theory. $\endgroup$
    – Arian
    Jun 15, 2021 at 23:21
  • $\begingroup$ For the CS theory describing the physical Hall effects $k$ does not have to be an ineger as the $A$ field is external rather than dynamical. $\endgroup$
    – mike stone
    Jun 16, 2021 at 11:42

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