You have just seen that Laughlin’s argument explains the quantization of the Hall conductance in terms of a pump which moves electrons through the bulk of a Hall cylinder, from one edge to the other of the cylinder.

Compare the situation with the simple electron pump which you studied earlier in the lecture. There, the pump moved electrons from one metallic lead to the other. Clearly the pump worked thanks to the availability of electronic states at the Fermi level in the two metallic leads. Otherwise, it would have no electrons to take and no place to drop them. Without the metallic leads, the pump would be like an empty carousel.

When applied to the Hall cylinder, this simple reasoning shows that Laughlin’s argument necessarily implies the presence of electronic states localized at the edges of the sample.

It is in fact very easy to convince ourselves that such states must exist. We just need to think again about the classical trajectory of an electron with velocity \(v\) moving in a perpendicular magnetic field \(B\). This trajectory is a circular orbit with radius given by the cyclotron radius.

What happens to the classical trajectory of an electron when the center of the orbit is too close to the edge of the cylinder, say closer than a cyclotron radius? It is easier drawn than said:

The electrons cannot exit the sample, and need to bounce back inside. This creates a so-called skipping orbit. In a real sample, there will be a confining electrostatic potential which keeps the electrons inside the Hall bar, or cylinder. The combination of a strong magnetic field in the bulk and a confining potential creates trajectories at the edges which are not closed, but travel along the full extent of the edges.

On the lower edge there are only left-moving states, and on the upper edge only right moving ones. On each edge there are only states moving in one direction, and the direction is opposite for opposite edges. These strange states obtained at the edges are often referred to as chiral edge states.

The chirality of the edges is determined by the orientation of the magnetic field (out of the plane vs. into the plane), and it would be reversed at both edges if the magnetic field were reversed.

The cartoon above is purely based on classical physics, and needs to be supplemented with quantum considerations before it can give quantitative predictions. We will soon see that the quantum version of the cartoon above can give an explanation of the quantized Hall effect, complementary to Laughlin’s argument.

But before we move on to that, we should realize that the picture above is in fact a manifestation of the bulk-boundary correspondence. Chiral edge states could not exist without the bulk of the quantum Hall sample.