Borrowing from Laughlin's argument on quantum Hall effect, Kane and Mele argued why there must be edge states in graphene with spin-orbit coupling in one paragraph, which is above the one with equation (6), of this paper:
C. L. Kane and E. J. Mele. “Quantum Spin Hall Effect in Graphene.” arXiv preprint cond-mat/0411737 (2004).
I do not understand this argument well and I hope someone can help. As I am aware of, there are proofs of the existence of the edge states. The most straightforward one is to solve the band dispersion, and another (popular) one is that the bands in the gapped bulk and the vacuum have different topologies so there should be a gapless edge state. Here I am just asking about the argument cited above.
As I understand, the authors are considering what happens if there is a voltage across the graphene sample. They imagine to shape the sample into a cylinder, and think the voltage to be induced by a slowly increasing magnetic flux inserted down the cylinder. I think these two steps are valid. For the first step, the reason is that the shape of the sample does not matter its properties too much provided sample is large enough, while for the second one, the reason is we can always think an electric field to be generated from the time dependent vector potential by $$\vec{E}=\frac{\partial\vec{A}}{\partial t}$$
But I do not understand their following argument: if a flux quanta is adiabatically inserted, a spin $\hbar$ is transferred from one end of the cylinder to the other, and there must be gapless states at each end to accommodate the extra spin. To make my question more clear, I do not understand
- Why is a spin $\hbar$ transferred from one end to of the cylinder to the other?
- Even if such a spin is transferred, why should there always be an edge state at each end?
- The authors emphasized this argument is valid only if $S_z$ is conserved, but what role does this conservation play in the argument?
I am confused because I am having trouble relating this argument to Laughlin's argument on integer quantum Hall effect. In Laughlin's argument, he considers the Hall bar as an annulus and the voltage as to be induced by inserting magnetic flux. However, Laughlin knows there are edge states at the edges where the energy levels are pushed up. Because of the existence of these edge states, upon sending a flux quanta $h/e$, by gauge invariance and explicit calculating of the Landau level wave functions, in each Landau level one state is "pushed from the bulk to the edge" and increases energy. Then one electron must go to the another edge to keep equilibrium, therefore, the transverse conductance is (roughly) \begin{split} \sigma_{xy}=\frac{I}{V}=\frac{ne/T}{h/eT}=\frac{ne^2}{h} \end{split} So my understanding of Laughlin's argument is that he used the existence and gauge invariance to calculate transverse conductance.
However, in Kane and Mele's argument, they are trying to prove the existence of the edge states instead of using the existence of edge states to calculate quantized spin conductance, so I do not see the clear relation between this argument and Laughlin's. It would be great if someone can point this relation out.