Let’s now try to obtain an alternative expression for the Hall conductance \(\sigma_H\) of our Hall bar. In general we expect the electric and magnetic fields present in our Hall bar to apply a force to the electrons, and increase their velocity.

Instead of solving the problem directly, let us make the ansatz that the electrons enter a state, which is obtained from the usual electron ground state by doing a Galilean transformation to a reference frame moving with velocity \(\bf{v}\) with respect to the original reference frame.

Since the average velocity of the electrons is \(\bf v\) in the original reference frame, the average force on the electrons is

\[{\bf F}= e\,(\mathbf{E}+\mathbf{v}\times \mathbf{B}).\]

If we want to be a steady state then \(\bf F=0\), which means that \({\bf v}= (\mathbf{E}\times \mathbf{B})/B^2\). Since the electrons move with an average velocity \(\bf v\), and if \(n\) denotes the electron density, we can easily guess that the current density is \({\bf j}=n e {\bf v}=(n e/ B) \,(\mathbf{E}\times \mathbf{z})\).

Comparing with the previous subsection, we can thus conclude that simply based on Galilean invariance, an electron gas in a magnetic field must have a Hall conductance that is given by

\[\sigma_H=n e B^{-1}.\]

This relation, which says that \(\sigma_H\propto n\), is extremely general in the sense that it does not depend on how the electrons interact with each other or anything else. It is referred to as the Streda relation. If we define the so-called “filling factor” as \(\nu=n h/ e B\) the Hall conductance can be written as a multiple of the quantum of conductance as \(\sigma_H=\nu \frac{e^2}{h}\).

As you already heard from Ady Stern in the intro video, people have measured the Hall conductance of this exact system to incredible precision. At relatively high density, the Hall conductance of this system behaves itself accordingly and scales linearly with gate voltage, which is tuned to control the density. At low filling factors, one would expect many non-idealities like disorder and interaction to break the Galilean invariance based argument and lead to a Hall conductance \(\sigma_H\) that varies from sample to sample and depends on disorder.