There is no really precise name for the 2D topological insulator with time-reversal symmetry. It is often called “\(\mathbb{Z}_2\) topological insulator.” However, this simply indicates that there are only two values of the topological invariant, and so it isn’t a very specific name.

The most commonly used name for this system is “quantum spin Hall insulator.” To understand why, let’s analyse a Hall bar made of such a non-trivial insulator. We will only need a Hall bar with four terminals, as shown below:

We have a finite voltage applied to terminal 1, so electrons are injected into the system from there. You can see that because of the helical edge states, there are as many modes connecting terminal 1 to terminal 3 as there are to terminal 4. A moment of thought, or otherwise a quick calculation, should convince you that in this case there is no net current flowing orthogonal to the applied voltage. The Hall conductance is zero, which is the expected result if time-reversal symmetry is preserved, as it is in our system.

However, counterpropagating edge states have to have exactly opposite spin due to Kramers degeneracy. This means that there may be a net spin current across the sample, orthogonal to the applied voltage.

In particular, let’s again make the simple assumption that the spin projection along some axis is conserved. Then, in the figure above, all modes colored in red have spin up, and all modes colored in blue have spin down. So terminal 1 distributes electrons coming out of it according to their spin: all electrons with spin up end up in terminal 4, and all those with spin down in terminal 3. The system has a quantized spin current between terminals 3 and 4, hence the name “quantum spin Hall effect”.

However, the quantized spin Hall current is not a general property of a quantum spin Hall insulator. Here, it arises because we have combined time reversal symmetry with a spin conservation law, and as we learned in the first week, conservation laws are boring from a topological point of view.