Theoretically, the hallmark of the topological insulator is the quantized conductance of the edge states that are protected from elastic backscattering. In the last unit, we learned that the key to this protection is time-reversal symmetry. Therefore, breaking time reversal symmetry by for example applying a magnetic field, should suppress the quantized conductance.

We can think about this more explicitly by considering a simple model for the helical edge states with a magnetic field \(\bf B\):

\[H=v_F k_x\sigma_z+{\bf B}\cdot {\bf \sigma},\]

where \(\bf \sigma\) are Pauli matrices representing the spin degree of freedom at the edge. This is what we get from the BHZ model, which conserves spin. For more general models we would interpret \(\bf \sigma\) as a pseudo-spin degree of freedom, which is odd under time-reversal.

If we consider the simple case of a magnetic field \({\bf B}=B {\bf x}\) along the x-direction, we find that the edge spectrum \(E=\pm\sqrt{v_F^2 k_x^2+B^2}\) becomes gapped. Clearly, the edge becomes insulating if we set the chemical potential at \(E=0\).

We can very easily calculate that this is the case if we plot the conductance of the QSHE model as a function of magnetic field:

However, even if we consider energies \(E>B\) above the gap, the eigenstates at \(\pm k_x\) are no longer Kramers’ pairs, i.e. related by time-reversal symmetry. Therefore, any mechanism which changes momentum by \(2 k_x\) can backscatter electrons from left movers to right movers.

Edges of semiconductors are typically quite disordered - so we expect the random potential at the edge to provide “elastic backscattering” that can change the momentum without changing the energy. Such backscattering, in addition to any other “inelastic backscattering” by phonons etc, would decrease the conductance of the edge from the ideal quantized value. If we set \(B=0\), elastic back-scattering that can occur at finite \(B\) is forbidden, so we generally expect the application of a magnetic field to reduce conductance of the edge.

We see below that indeed the conductance of the \(L=20\,\mu m\) device is strongly reduced by the application of a magnetic field:

(copyright JPS, see license in the beginning of the chapter)

However, we notice that this effect seems to work only when the magnetic field is perpendicular to the sample. In-plane magnetic fields do not seem to do a whole lot (there is an effect, but much larger fields are required). According to our model Hamiltonian, an in-plane field should have opened a gap, while a perpendicular field which adds a term proportional to \(\sigma_z\) should have not done anything.

So, while the experiment sees something similar to what we hoped to find using a simple theory, the effect of the magnetic field seems reversed. There may be several explanations for this phenomenon, such as a presence of extra terms in the Hamiltonian that rotate the spins of the edge states without breaking time-reversal symmetry.

You might be worried that the suppression of conductance is only shown for the long device, which does not show quantized conductance. If you are, then you are absolutely right in worrying about this :-)

Localization of QSHE edge states by magnetic field is relatively poorly understood, and we are not aware of a final experiment that would prove its existence or tell us in details what it is that happens at the QSHE edge in a magnetic field. As you will learn in two weeks, opening the gap by magnetic field opens new pathways for the creation of Majoranas, and so it is still a very important direction of research.