# The science of going in circles on doughnuts

The paper that I’m going to be reviewing today is A Topological Look at the Quantum Hall Effect. It explains an amazing coming together of topology and quantum physics, of all things, and provides a review of very important, Nobel prize winning work in Physics.

## The Hall Effect

We all learned in high school that if we pass current through a conductor in a magnetic field, and the magnetic field is not parallel to the current, then the conductor experiences a force.

But does the conductor as a whole experience this force, or only the electrons inside it? Maxwell, in his Treatise on Electricity and Magnetism, said that “the mechanical force which urges the conductor…acts not on the electric current, but on the conductor which carries it.” This seems wrong, right? It is the charges that should experience the force, and not non-charged portions of the conductor.

Edwin Hall, a student at Johns Hopkins University, investigated further. He passed a current through a gold leaf in a magnetic field. He could see that the magnetic field was acting on the electrons themselves, as it altered the charge distribution inside the gold leaf, which he could measure with a galvanometer.

As you can see, there is a separation of charges across the breadth of the plate. One may imagine that instead of flowing through the whole plate uniformly, as soon as the current enters the plate, it gets deflected to one side of the plate, although it keeps moving forward. The potential difference created by this separation of charges is known as Hall voltage. Hall conductance is the current already flowing through the plate, divided by the hall voltage. Remember Ohm’s law, which says $V=IR$? This implies that $\frac{1}{R}=\frac{I}{V}$. This $\frac{1}{R}$ is the conductance. Of course the difference is that this current is not caused by the Hall voltage. Hence, this formula cannot directly be obtained from Ohm’s law, but let’s shut our eyes to these details.

The direction of this voltage is not fixed, and depends on the charge of the charge carriers in the conductor. Hence, measuring the direction of this voltage is an easy way to determine the nature of charge carriers in a conductor. Some semiconductors owe their efficacy to having positively charged charge carriers, instead of the usual electrons.

## The Quantum Hall Effect

In 1980, Klaus von Klitzing was studying the conductance of two dimensional electron gas at very low temperatures. Now remember that $resistance=voltage/current$ (because $conductance=current/voltage$ as discussed previously). As voltage increases, this formula says that resistance should increase. Now as the Hall voltage increases with an increase in magnetic field strength, as the magnetic field strength increases, so should the resistance. But what is the pattern of this increase? It is pretty surprising, surprising enough to get Klitzing a Nobel prize for studying it.

Note: it seems that the resistance is increasing in small steps for lower magnetic field strength, but bigger steps for higher values of the magnetic field. However, the conductance, which is its reciprocal, decreases by the same amount for each step. In other words, conductance is quantized. A quantity if referred to as quantized if can be written as some integer multiple of some fundamental quantity, and increases or decreases by the steps of the same size.

Why do we get this staircase graph, and not, say, a linear graph? If we consider the conductance=1/resistance values in the above graph, we see that all the successive values are integer multiples of a constant we find in nature- $e^2/h$, irrespective of the geometric properties or imperfections of the material. Here $e$ is the electric charge of an electron, and $h$ is Planck’s constant. Why does this happen?

Robert Laughlin offered an explanation. Consider an electron gas that is cold enough, such that quantum coherence holds. This basically means that there’s not much energy in the system for all the particles to behave independently, and that all particles behave “similarly”. Hence, the behavior of the system can be described easily by a Hamiltonian dependent on a small number of variables ($2$ in this case). Laughlin imagined the electron gas to be a looped ribbon, with its opposite edges connected to different electron reservoirs.

Now imagine that there’s a current flowing in the ribbon along its surface in a circular motion. The presence of the magnetic field causes a separation of charges. Because the two edges are already connected to electron reservoirs (which can both accept and donate electrons easily), the separation of charges causes electrons to flow from one reservoir to another. If, say, the side connected to $A$ becomes positive due to separation of charges and the side connected to $B$ becomes negative, then electrons from $A$ go into the ribbon, and electrons from the ribbon go into $B$. One may imagine a current flowing between the electron reservoirs, which is maintained because the magnetic field maintains the Hall voltage between the two ends of the ribbon. However, in order to increase this Hall current, the magnetic flux $\Phi$ will have to be increased (which will increase the Hall voltage).

The Aharonov-Bohm principle says that the Hamiltonian describing the system is gauge invariant under (magnetic) flux changes of $\Phi_0=hc/e$, which is the elementary quantum of magnetic flux. The Hamiltonian of a system may be thought of as its “energy”. Hence, although the Hall voltage increases every time we increase the magnetic flux by one quantum, the “energy” of the system remains unchanged. This quantized increase in Hall voltage causes a quantized increase in resistance, which can be seen from the step graph above. If the increase in resistance was not quantized, then the graph would have been a smooth curve, and not “stair-like”.

A small note about what it means for the Hamiltonian to be gauge invariant under flux changes of $\Phi_0$: it is not like the Hamiltonian doesn’t change with time. By the time the flux increase happens, all he eigenstates of the Hamiltonian will have changed. What gauge invariance here means is that if the measured eigenstate of the Hamiltonian at time $t_0$ is $\psi(t_0)$, and we complete increasing the the magnetic flux by $\Phi_0$ at time $t_1$ and also measure the Hamiltonian at that time, then the eigenstate we get is going to be $\psi(t_1)$, and not some other eigenstate $\psi^*(t_1)$. For simplicity, we will just assume that $\psi(t_0)=\psi(t_1)$ in this article, and that the Hamiltonian doesn’t change with time.

Another important note is the following: the charge transfer between the two reservoirs is quantized. In other words, when I increase the magnetic flux by one quantum of flux, the Hall current increases by a certain amount. When I increase the magnetic flux by the same amount again, the same increase in Hall current is seen. There is no good reason for this in quantum mechanics. This is because although the increase of the magnetic flux by $\Phi_0$ brings back the system to its original Hamiltonian eigenstate, in quantum mechanics there is no guarantee that the same initial conditions will lead to the same outcome. Quantum states randomly collapse into any one of their eigenstates, and don’t have to collapse to the same eigenstate each time. This fact, of the same increase in Hall current, needs some topology to be explained. But before that, we explore some concepts from geometry- namely, curvature.

“Adiabatic” evolution occurs when the external conditions of the system change gradually, allowing the system to adapt its configuration.

What is curvature? Let us explain this with an example. Take the earth. Starting at the North Pole, build a small road that comes back to the North Pole. By small, we mean that the road should not go all the way around the earth. If you drive your car along this road, the initial orientation of the car will be different from the final orientation of the car (when you arrive). Curvature is the limit of this difference between initial and final orientations of the car, divided by the area enclosed by the road, as the loop tends to a point. Note that this would never happen on flat land which was “not curved”: in other words, the initial and final orientations of the car would be the same.

But how does curvature fit into quantum mechanics? Consider a curved surface parametrized by two angles- $\theta$ and $\Phi$. Each point of this curved surface corresponds to an eigenstate of the hamiltonian, which can be written as $H(\theta\Phi)$. Essentially, this is a configuration space of the Hamiltonian of the system. Let us now build a quantum (ground) state, say $e^{i\alpha}|\psi(\theta,\Phi)$. If we build a small loop from a point to itself, this ground state, as it is “driven” around the loop, changes its phase from $\alpha$ to something else. The curvature of this configuration space, measured using the limiting process as described above, turns out to be $2 Im \langle \partial_{\theta}\psi|\partial_{\Phi}\psi\rangle$.

Why do we care about bringing things back in a loop, as far as Hamiltonians are concerned? This is because at the end of the process of increasing the magnetic flux by one quantum, the initial and final Hamiltonian has to be the “same” in our case. We have to always come back in a loop to our initial Hamiltonian.

## Hall Conductance as Curvature

What are $\theta$ and $\Phi$ as described above? $\Phi$ is associated with the Hall voltage, and $\theta$ can be thought of as a function of $\Phi$. More precisely, $theta$ is defined such that $I=c\partial_{\theta}H(\Phi,\theta)$. The Hamiltonian is periodic with respect to $\Phi$ or Hall voltage. If the period of $H$ with respect to $\Phi$ is $P$, then the process of increasing the magnetic flux by one quantum can be thought of as changing the value of $\Phi$ to $\Phi+P$, which leaves the Hamiltonian unchanged. Geometrically, if the configuration space can be thought of as a curved surface, this process can be thought of as moving along a circle and returning to the point we started out from.

For an adiabatic process where $\Phi$ is changed very slowly, Schrodinger’s equation gives us $\langle \psi|I|\psi\rangle=\hbar cK\Phi$. $\langle \psi|I|\psi\rangle$ can be thought of as the expected value of the current (the weighted average of all possible values of the current), and we know that $\Phi$ is voltage. Hence, $K$, going by the formula discussed above, is (some constant multiplied with) conductance. In this way, Hall conductance can be interpreted as the curvature of the configuration space of the Hamiltonian.

## Chern numbers

What is this “curved surface” that forms the configuration space of the Hamiltonian? As it is periodic in both its parameters $\Phi,\theta$, we can think of it as a torus. Each point of this configuration space corresponds to an eigenstate of the Hamiltonian.

Let the torus given above be the configuration space of the eigenstates of the Hamiltonian. Consider the integral $\int K dA$. The difference between this integral evaluated on the orange patch and the external blue patch is an integral multiple of $2\pi$. This integer is known as Chern number. You may have seen the famous Gauss-Bonnet Theorem, which states that $\frac{1}{2\pi} \int K dA=2(1-g)$. The right hand side, which is $2(1-g)$, is the Chern number for in the special case that the loop around the orange area is contractible (can be continuously shrunk to a point).

The Chern number associated with the diagram given above does not change when we deform the orange patch slightly. However, it will change if the boundary of the orange patch changes its topological properties (i.e. if its homotopy group changes). One way it could do that is if the loop circled the “doughnut-hole” of the torus completely.

## Let’s bring it all together

Now that we have all that seemingly useless background, let’s bring it all together. Let’s start with the torus, representing all eigenstates of the Hamiltonian. Every time we increase the magnetic flux by one quantum, it’s like we’re going one full circle around the doughnut, and returning to our original Hamiltonian eigenstate. Both the pink and red curves given below are examples of this.

This causes the Chern number to change by an amount, say $a$. Why does the Chern number change? Because we’re adding the integral $\frac{1}{2\pi}\int K dA$ to the previous integral. When we again increase the magnetic flux by one quantum, we go in another full circle around the torus. This causes the Chern number to change by the same amount $a$. Both the charge transfer between the conductance are exactly this Chern number in fundamental units (where constants like $h$ and $c$ are taken to be $1$). Hence, each time we increase the magnetic flux by one quantum, we see the increase in Hall current, and the same decrease in conductance.

Why does topology have anything to do with quantum mechanics? We might think of this as an analogy that we constructed, that just happened to work absurdly well. We interpreted conductance in terms of curvature. Hence, the change in conductance can be thought of as an integral of curvature on copies of the torus. These integrals satisfy strict mathematical relations on the torus, via the Chern-Gauss-Bonnet Theorem. Hence, we can use this theorem to approximate how much the conductance changes every time we move along a certain loop on the torus (every time we increase the magnetic flux by one magnetic flux quantum).