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

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title: 4. The Quantum Hall Effect (conflicted)
ctime: 20220517090956
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# 4. The Quantum Hall Effect


{/** TODO: Move the Quantum Hall Effect section to a better spot
    *
    * labels: order, III
    * milestones:
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- Following @Norton2012, we know how to interpret this: we can demote an idealization to an approximation if we are able to show that there is a
    smooth limit.
- This is exactly the problem, according to @shech2019 the topological explanation of anyons requires an _exact_ 2d system (or a rather ad-hoc 2d system as we saw).
- Therefore we cannot call this approximately 2d.
- According to @shech2019, this is a problem for those with nominalist inclinations,[^1] as the instantiation of abstract mathematical structures in the real world would be an issue.
- Such an argument can only be made, however, through the use of the eia which was discussed in the previous (or next?) chapter.
- Therefore, it would do such people well to examine the physical effect thoroughly for any signs of this.

In this chapter, we will examine the inner working of the Quantum Hall Effect. This is needed in order to understand the following

1) why do people say that these particles are anyons
2) what idealizations are made in the explanation of the Quantum Hall Effect separate from the topological one.
3) to calculate the berry phase

Readers less interested in the details can skip to the last subsection of this section `link` and the next section `link`.


{/** TODO: Justify the connection between the quantum and classical hall effect slightly better
    * justify this slightly better
    * labels: expansion
    * milestones:
    */}


The Quantum Hall Effect is the quantum version of the classical hall effect, a rather straightforward consequence of moving charged particles in a magnetic field. It was discovered all the way back in `xxxx` by edwin hall, who claimed it a novelty which would never have any use-case.


{/** TODO: Add source about the Quantum Hall Effect
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Today hall-effect sensors are widely used to pick up magnetic fields and rotations of conducting materials. More esoteric applications are as potential long-term space-travel propellants in so-called hall-effect thrusters, which are a specific form of ion propolsion, which can generate a low thrust for a very long amount of time.


{/** TODO: Rewrite the basics of the Classical Hall Effect in words, not bullet points
   * 
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The basics of the effect are rather simple

- strip which has current flowing through it.
- magnetic field perpendicular to that strip
- induces lorentz force which pushes particles to one side, increasing their concentration and creating a potential difference, inducing a voltage changing the resistivity.
- this resistivity linearly depends on the voltage, which linearly depends on the concentration difference which linearly depends on the strength of the magnetic field, thus yielding a linear dependence of the resistivity to the strength of the perpendicular magnetic field.
- looks like figure ref:fig:che

However, as most classical effects, they only hold in a certain regime. When we clean up the materials, lower the temperature dramatically ($<2k$) and increase the magnetic field strength considerably, we observe the _Quantum Hall Effect_ (QHE): rather than a linear dependence on the magnetic field strength, the resistivity shows these strange plateaux at rather consistent values, as inref:fig:QHE. The spacing between these plateaux does not seem to differ between different materials, which is even more curious.

So: why? why do we see these plateaux, and why at these levels? the short answer (partially) is: the impurities in the sample lift the degeneracy of the eigenstates of the Quantum Hall system, called landau levels, and _localizes_ some of those states, rather than the states extending from one edge of the sample to the other (akin to the wavefunctions of a particle in a box). These localized states do not conduct current (as do not span the system, and resistivity is measured from end to end), thus leading to the plateaux in the resistivity: we change plateaux when all the localized states are filled and we
Move to different extended states. The specific values of these levels depend on weird physics at the boundaries of the system called _edge-modes_, but in short the levels correspond to the number of filled landau levels: at higher magnetic field strengths the levels can accommodate more states, thus the lower the magnetic field strength the larger the number of available levels, leading to a smooth-looking linear dependence in the classical regime.

There are quite some caveats to the above story, but that is the gist of why the effect happens for _integer levels_. The less interested reader can skip over the following section describing the corresponding _integer Quantum Hall Effect_ (IQHE) and head to the description of its fractional cousin, where the same does not hold. Specifically, the IQHE requires one massive idealization: electron interaction is neglected completely. As we shall see, this idealization cannot hold in the FQHE.

## The Integer Quantum Hall Effect

In order to do as little physics as is possibly required, i will skip most of the justification for the quantum formalism of the QHE (quantizing the classical hamiltonian, finding the commutators) and many of the intermediate steps to arrive at the relevant results. I point the interested reader to @tong2016 for a quite accessible and to @arovas2020 for a more thorough pedagogical discussion of these issues, and to @stone1992, @prange1987, @doucot2005 for rather complete, but less pedagogical sources.

As mentioned previously, the integer Quantum Hall Effect (IQHE) is the observation of plateaux in the hall resistivity $\rho_{xy}$ at regular intervals. These intervals happen to be integer multiples of the _quantum of resistance_  $r_q=\frac{2\pi\hbar}{e^2}$. `check whether this is so`

- this is rather curious, why so precise?
- in fact, so precise that these experiments are used to determine the quantum of resistance.
- why are they there, and why at those levels?

As physicists, the first thought as to the origin of these plateaux probably goes to the energy eigenstates of the system, which turns out to be correct! the plateaux correspond to the general energy eigenstates of a system of charged particles moving in a perpendicular magnetic field, so called landau levels.

- it will be difficult to properly understand the QHE without first briefly going over landau levels.

`assumptions`

### Landau Levels

Here we already start to sneak in our idealizations: we will treat this system as if it is two dimensional. `however, this system can be easily extended to 3d, see xxxx`

The simplest hamiltonian for a system of particles moving in a magnetic field is

$$h=\frac{1}{2} m(\hat{\mathbf{p}} +e \hat{\mathbf{a}} )^2$$

The magnetic field is perpendicular to the $x,y$-plane, so we define the vector potential $\hat{\mathbf{a}}$ using our knowledge that the magnetic field is perpendicular to the plane $\nabla\times\hat{\mathbf{a}}=b \hat{z}$, to be

$$
\hat{\mathbf{a}}=\begin{pmatrix}
0\\

Xb\\

0
\end{pmatrix}
$$

`this will be appendix` the easiest way to find the energy eigenstates is the way all hamiltonians get solved: treat is as the harmonic oscillator. As the hamiltonian of the "normal" harmonic oscillator is


{/** TODO: Add Hamiltonian for generic harmonic Oscillator
  * 
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These allow us to define raising and lowering operators`at this point we introduce new variables. These are raising and lowering operators, entirely analogous to those that we use in the harmonic oscillator. They are defined by`$a= \frac{1}{\sqrt{2e\hbar b}} (\pi_x - i\pi_y) \quad a^\dagger = 1\frac{1}{\sqrt{2e\hbar b}} (\pi_x + i\pi_y)$

The commutation relations for $\pi$ then tell us that $a$ and $a^\dagger$ obey
$$
[a, a^\dagger] = 1
$$

$h= \frac{1}{2}m \pi^2   = \omega_b \left( a^\dagger a + \frac{1}{2}  \right)$

We find that the energy eigenlevels are $\ket{n}=\omega_b\left(n+\frac{1}{2}\right)$

### Calculating the Degeneracy

We started by saying that the plateaux in the IQHE correspond to the various filled landau levels of the simple particle in a magnetic field system. We still need to prove this, namely by deriving the conductivity for the hall states.

This can get rather messy, so i will show put the derivation for a single particle here, and the more general derivation of the so called _kubo formula_ in the appendix.

To find the resistivity, we use ohm's law, which relates the energy of a particle to the current density (current over area)

$$\mathbf{e}=\sigma \mathbf{j}$$

Our mechanical momentum is$\hat{\mathbf{\pi}}=\hat{\mathbf{p}}+e \hat{\mathbf{a}} = m \hat{\mathbf{\dot{x}}}$

Classically, the current (for a single particle) is simply $\mathbf{i}=-e\mathbf{\dot{x}}$, but isnce we are working quantum mechanically we take the expectation value

$i=-e/m \sum_filled_states \bra{\psi}-i\hbar\nabla+e \hat{\mathbf{a}}\ket{\psi}$

We are working in landau gauge.

…

We end up with

### Edge Modes

In order to provide a more thorough calculation (i.e. Not just considering a single electron) we take advantage of a the fact that the system is bounded. I will not reproduce this here.

### Robutsness

The explanation above shows us why there are plateaux at the levels we see, and even gives us a hint as to why those states would be rather stable, but it has not yet told us _why_ these plateaux persist over a range of values yet, just that something is going on at those values. We have only shown that at complete filled landau levels ($\nu\in \mathbb{n}$) the longitudinal resistivity $\rho_{xx}=0$ and that the transversal resistivity $\rho_{xy}$ is an integer multiple of the quantum of resistance. However, if we were to move even slightly away from the completely filled landau state where $b=\frac{ne}{2\pi\hbar}\frac{1}{\nu}$, all our previous arguments hold no water and there is no reason to expect anything already covered to hold.

This is obviously a problem, as an effect which only shows up at a specific real number would never be experimentally observable. We will need do some dirty work in order for `this` to make sense. In fact, the solution not only requires some dirty work: the solution _is_ dirtiness.

Experimental samples are inherently dirty (here meaning: containing other elements than the intended sample[^2]), and these impurities require us to re-examine our previous claims somewhat.[^3] these impurities lead to two vital insights which will allow us to solve our puzzle:

1) they (unsurprisingly) break the degeneracy of the landau levels, resulting in more swept out states as in ref:fig:disorder
2) they (more surprisingly) turn many _extended_ quantum states into _localized_ ones.

![](../../media/broadlandau.png "density of states in the IQHE with and without disorder")

![](../../media/breakdegen.png "extended to localized states")

This might sound all well and good, but certainly there is a limit to the amount of disorder we are allowed to introduce into our system? surely the spokes of my bicycle should not be able to serve as Quantum Hall systems.

Correct you are: in general we demand that a) the strength of the disorder

(which we model as a random potential) ought to be small relative to the landau level splitting and b) the disorder does not dramatically vary on small scales,

Such that for a particle influenced by it the potential can locally be seen as constant. We can express these as

$$v_{disorder}<<\hbar\omega_b$$

And

$$|\delta v|<<\frac{\hbar \omega_b}{l_b}$$

Where $l_b$ is the magnetic length, `roughly the length scale at which these effects are relevant`


{/** TODO: Rewrite this copy pasted section by Tong in your own words.
    * Now consider what this means in a random potential with various peaks and troughs. We’ve drawn some contour lines of such a potential in the left-hand figure, with + denoting the local maxima of the potential and − denoting the local minima. The particles move anti-clockwise around the maxima and clockwise around the minima. In both cases, the particles are trapped close to the extrema. They can’t move throughout the sample. In fact, equipotentials which stretch from one side of a sample to another are relatively rare. One place that they’re guaranteed to exist is on the edge of the sample.
    The upshot of this is that the states at the far edge of a band — either of high or low energy — are localised. Only the states close to the centre of the band will be extended. This means that the density of states looks schematically something like the right-hand figure.
    Conductivity revisited for conductivity, the distinction between localised and extended states is an important one. _Only the extended states can transport charge from one side of the sample to the other. So only these states can contribute to the conductivity._ Let’s now see what kind of behaviour we expect for the conductivity. Suppose that we’ve filled all the extended states in a given landau level and consider what happens as we decrease b with fixed n. Each landau level can accommodate fewer electrons.
    But, rather than jumping up to the next landau level, we now begin to populate the localised states. Since these states can’t contribute to the current, the conductivity doesn’t change. This leads to exactly the kind of plateaux that are observed, with constant conductivities over a range of magnetic field. So the presence of disorder explains the presence of plateaux. But now we have to revisit our original argument of why the resistivities take the specific quantised values (2.3). These were computed assuming that all states in the landau level contribute to the current. Now we know that many of these states are localised by impurities and don’t transport charge. Surely we expect the value of the resistivity to be different. Right? well, no. Remarkably, current carried by the extended states increases to compensate for the lack of current transported by localised states. This ensures that the resistivity remains quantised as (2.3) despite the presence of disorder.
    * labels: rewrite
    */}


I will leave the more detailed explanation for why the extended states compensate for the localized states for the interested reader, see text [@Tong2016].

That is it for the IQHE, as we derived the two things we needed from it: we gained a general understanding of why the plateaux have their values (they are the energy eigenvalues of the landau levels) and, more importantly, we roughly understand why these plateaux are robust. Unfortunately, this is not the end of the story, we did not even mention anything topological yet! For that we finally turn to the fractional Quantum Hall Effect.

## The Fractional Quantum Hall Effect

Following the naming convention of the IQHE, the Fractional Quantum Hall Effect (FQHE) refers to the observation of plateaux at _fractional_ values of the quantum of resistance in the hall resistivity $\rho_{xy}$. Sadly, very few of the arguments mentioned above will be able to explain these plateaux, as we have only shown that they appear at fully filled landau levels. However, our intuition about the robustness will still hold.

The goal of this exposé is twofold. First we want to gain a general understanding of how the FQHE is thought about in general: what assumptions go into calculating the relevant parameters, which idealizations are noteworthy,

Etc. The other main goal is the 'derivation' of the Laughlin wavefunction, the wavefunction used to describe the fqh system, and its excitations. At the end we will arrive at the problem of calculating the exchange statistics of these excitations, which turn out to be _anyons_, but not actually compute them yet:

This will be done in section ref:sec:geometricphase.

The key difference between the description of the fractional as opposed to the integer Quantum Hall Effect is the inclusion of electron interactions in the former, which becomes impossible to ignore at the energy scales above $\nu=1$,

Which is where most of the FQHE physics is done. As a result the reasoning cannot be as rigorous as before. In the IQHE we could pretend that all the states occupied the same landau level, which allowed us to calculate the wavefunctions and energy levels, leading to the derivation of the hall resistivity and confirm our suspicion that the plateaux correspond to fully filled landau levels. Only after doing that did we let go of that idealization and allow the degeneracy to be lifted in order to argue that the plateaux were robust.

This order of operation is no longer possible in the FQHE, as the electron interactions lift the degeneracy of the landau levels from the start, forcing us to compute the wavefunctions in a different way. A first approach would be to use perturbation theory: model the electron interaction as a small perturbation to ref:eq:landauham and then gradually compute a better and better approximation to the actual wavefunction. While this is fine for simple two-electron systems,

The number of electrons in a qh system is closer to $10^{23}$. That is a rather large matrix to diagonalize, not even close to possible to do numerically.

Therefore, we need to pull some tricks.

The trick is: do not compute the wave function, just write one down. That is exactly what text [@Laughlin1983] did, yielding what we now call [[the Laughlin wavefunction]].

Laughlin of course did not just simply write down a bunch of wavefunction and pick the one he liked best, it is motivated by some observations from the system and from some general conditions we have to place on any wavefunction.

Specifically, in a previous paper [@Laughlin1983a] he derived the wavefunctions for three particles in the FQHE. Recapping this in the case of two electrons is worthwhile.

We have a system of two electrons with a potential $v(|r_1-r_2|)$.

- to solve such systems, it's easiest to work with angular momentum.
- if we want to work with angular momentum, the gauge we picked before (landau gauge) is not very useful, as it does not include any kind of rotation

  - 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.