Why does the WF being in the lowest landau level entail that f(z) must be analyt...

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https://github.com/ThomasFKJorna/thesis-writing/blob/5f1acf2c13d516eabf341771a257f2f66d67c8a1/Chapters/III. Anyons/4. The Quantum Hall Effect (conflicted).md#L234

  - instead we pick _symmetric gauge_, which is $\hat{\mathbf{a}}=-\frac{1}{2}\hat{\mathbf{r}}\times \mathbf{b}= -yb/2 \hat{\mathbf{x}} + xb/2 \hat{\mathbf{y}}$
- skipping multiple steps ahead, we see that the wavefunctions look like

$$
       \psi \∼ (z_1 + z_2)^m (z_1 − z_2)^m \exp{ − \frac{(|z_1|^2+|z_2|^2)}{4l^2 b}}
$$

Unfortunately this does not uniquely or straightforwardly generalize to $N$ particles, so we have to pull a few more tricks.

1) we do try to generalize, and say that a wavefunction for n particles will look something like $f(z)e^{something}$
2) we _insist_ that the wavefunction be in the lowest landau level (lll) it can possibly be: there are no other landau levels it can fall back on.
{/** TODO: Why does the WF being in the lowest landau level entail that f(z) must be analytic?
  * 
  * labels: expansion
  **/}
         this is rather strong. This leads to the claim that $f(z)$ _must_ be analytic. 
3) since the state will have to describe fermions, the wave function must be anti-symmetric under exchange of the particles, which requires $f(z)$ to be odd.
4) finally, since we need to conserve angular momentum, we require that `f(z) be a homogeneous polynomial of degree m, where m is the total angular momentum.`

All these constraints add up to the fact that $f(z)=\prod_{j<k}(z_j-z_k)^m$, with $m$ odd.==this is what people mean when they say that the LWF falls into the same universality class as the actual wavefunction, as any wavefunction needs to account for this.== [^4]

Well, neat, you might say, but this still just describes fermions, i thought we were going to be talking about anyons! right you are, things only really get exciting once we start talking about _excitations_ of this ground state.

{/** TODO: Rewrite Laughlin's description about "piercing" flux in actual people terms
     * I generate the elementary excitations of g by piercing the fluid at z, with an infinitely thin solenoid and passing through it a flux quantum t) cp =- hc/e adiabatically. The effect of this operation on the single-body wave functions is (z-z, ) exp(-4lzl')-(z-z, ) "exp(--'. ~z~'). (») let us take as approximate representations of these excited states (13) '4 "=&.,''4"(--'xl, (, l')in(, ——". I(n,.(*, —, )"),
     * labels: rewrite
     **/}

- the excitations of the FQH state (technically of the state described by the Laughlin wavefunction, which are not exactly alike)

# Footnotes

[^1]: The non-dirty sample just discussed is a great example of taking an idealization too seriously: we cannot explain the phenomena in the idealized setting, and have to retrofit extra physics on top of the idealization in order to get a satisfying explanation, only for us to then show in the limit of no impurities we regain our initial idealization. If this limit would not have been smooth (it fortunately is), we would have wasted all this time!
[^2]: Improved performance of impure 2D materials is an active area of research in material science, see [@Wang2020] for a summary of how impurities enhance the conductivity of graphene.
[^3]: The LWF might appear to just be an easy to compute with exemplar of this universality class, but it actually has some other nice features which set it apart from its siblings, see `other nice feature of LWF`
[^4]: Scare quotes, as the derivation clearly does not rely on experimental measurement alone. Unfortunately it is not possible to simply measure the number of electrons in a given area.
[^5]: If the reader was not aware of this yet, the author does hold such inclinations.