Why is the quantized Hall conductance \(\sigma_H\) so robust and independent of system details? Clearly there must be a topological argument at play.

Soon after the experimental discovery, Laughlin came up with an elegant argument that mapped the Hall conductance problem to a topological pumping problem and in the process explained the robustness. Let us go through this argument.

The Corbino geometry

To start with, we imagine doing the Hall measurement in a system cut out as an annulus, which is referred to as the Corbino disk:

We will also try to do the experiment in reverse i.e. apply an electric field along the circumference of the disk and measure the current \(I\) in the radial direction, as shown in the figure. The radial current is easy to measure - we just measure the amount of charge \(\Delta Q\) transferred between the inner and outer edges of the Corbino geometry and obtain the radial current \(I=\Delta Q/\Delta T\), where \(\Delta T\) is the time over which this is done.

But how do we apply an electric field in the tangential direction? The easiest way to do this is to apply a time-dependent magnetic field in the centre of the disc and use the Faraday effect.

We can calculate the electric field from the changing magnetic field using Faraday’s law as \(\oint d{\bf{r}\cdot\bf{E}}=\partial_t \Phi\), where \(\Phi\) is the magnetic flux resulting from the field in the center of the disk. Assuming that the electric field depends only on the radius \(R\) we find that the resulting tangential electric field is given by

\[E(R,t)=\frac{1}{2\pi R}\,\partial_t \Phi.\]

Given \(I\), we can also calculate the other component of the measurement of the Hall conductance \(\sigma_H\) i.e. the radial current density \(j=I/(2\pi R)\) at the same radius \(R\) as we calculated the electric field.

Now that we know both the circumferential electric field and also the radial current density, the Hall conductance can be measured easily in this geometry as

\[\sigma_H=\frac{j}{E(r,t)}=\frac{I}{\partial_t \Phi}.\]

You might worry that we were a bit simplistic and ignored the longitudinal conductance in this geometry. We could measure the longitudinal conductivity by applying a voltage difference between the inner and outer edges and measuring the resulting radial current \(I\). For the remainder of this discussion, we assume that the longitudinal conductivity vanishes as is observed experimentally.

Laughlin pump

We are now ready to present the pumping argument to explain why the low temperature Hall effect is quantized.

To do this, we change the magnetic field in the center of the Corbino disc so that the flux changes by \(\Delta \Phi=\Phi_0=h/e\), i.e. a flux quantum over the time \(\Delta T\). (Note that this flux quantum is only half of the superconducting flux quantum that we were using last week. That’s because now the current is being carried by electrons and not Cooper pairs. It is customary to use the same symbol \(\Phi_0\) for both, since they often appear in different contexts). Assuming that we have a system with Hall conductance \(\sigma_H\), we obtain the charge transferred as

\[\Delta Q=I \Delta T=\sigma_H\, \Delta T\, \partial_t\Phi =\sigma_H\,\Delta\Phi=\sigma_H\, \frac{h}{e}.\]

Writing \(\sigma_H=\nu e^2/h\), we obtain \(\Delta Q=\nu e\). Since the longitudinal conductance \(\sigma_L=0\), we expect the system to be gapped in the bulk of the disc and we expect the entire charge transfer \(\Delta Q\) to occur between the edges.

Since the flux \(\Phi\) in the center is a flux quantum \(\Phi_0\), the wave functions of the electrons all return to being the same as at \(\Phi=0\). Therefore only an integer number of charges \(\Delta Q=n e\) can be pumped between the edges. This is Laughlin’s argument for why the Hall conductance must be quantized as

\[\sigma_{xy}=n e^2/h.\]

What you notice at this point is that we basically have a pump similar to the last unit.

Here an integer number of charges is pumped from one edge to the other as the flux \(\Phi\) is increased by \(\Phi_0\). As one sees below, one can simulate electrons in a Corbino geometry and check that indeed an integer number of charges is pumped between the edges as the flux \(\Phi\) is changed by \(\Phi_0\).