Kane and Mele's argument on the existence of edge states in quantum spin Hall effect of graphene

Question

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

1. Why is a spin $\hbar$ transferred from one end to of the cylinder to the other?
2. Even if such a spin is transferred, why should there always be an edge state at each end?
3. 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.

Answer 1

All three questions can be answered by first artificially separating the graphene sheet into two sheets:

• (a) first sheet with only spin up electrons, and
• (b) second sheet with only spin down electrons.

This statement alone should partially answer your third question; for the sake of organization, however, I will repeat a summary of this paragraph (in the end) anyway. This step of artificially separating spin species cannot be done unless $s_{z}$ is conserved. Spin-orbit coupling can be interpreted as a form of “spin scattering” which couples states with different spin. If different spin states are not decoupled then decoupling the sheet into (a) and (b) would not faithfully represent the original system. Hence conservation of $s_{z}$ is a necessary condition.

Now, according to the last paragraph of the left column (same page), the authors (indirectly) say that these two sheets independently realize Haldane’s model for spinless electrons; this is nothing but a lattice realization of the quantum Hall effect with zero net magnetic field. We can now apply Laughlin’s argument to the two sheets independently. There is, however, one thing to watch out for: the signs of the gaps for the spin up ($s_{z}=+1$) and down ($s_{z}=-1$) electrons are opposite. Note: in Eq. (3) you will either get $\pm \Delta_{{\rm so}}$ ($s_{z}=\pm 1$). Hence the transverse pumping of spins will occur in opposite directions for spin up and down electrons. Kane and Mele say the same thing (in different words) just a few lines above Eq. (5). Consequently, an up spin of $\hbar/2$ is pumped from (say) edge 1 to edge 2 for sheet (a) and a down spin of $\hbar/2$ is pumped from edge 2 to edge 1 for sheet (b). Hence a net spin of $\hbar$ is pumped from one edge to the other regardless of which you choose to label as “up” or “down” (or 1 or 2). Note: $\lambda_{R}$ is still assumed to be zero. That should answer your first question.

Note that in the paragraph above Eq. (6) the authors say “...adiabatically insert a quantum $\phi=h/e$ of magnetic flux quantum down the cylinder (slower than $\Delta_{{\rm so}}/\hbar$).” This means that the longitudinal electric field does not impart enough energy, to an electron in the highest occupied Landau level, such that it can overcome the mobility gap (in the case of the integer quantum Hall effect). Hence the only way a state is available for the pumped electron (or spin), on the other edge, is if it had sub-gap states. In other words, the edges are gapless.

I apologize for messing up the order of the questions; my explanation required this order (no pun intended). Anyways, here’s a summary:

1. The pumping of spins can be explained by using the same gauge invariance in the Laughlin argument. This is much easier to see once you split your system into two spinless systems with each experiencing opposite effective magnetic fields.
2. A system with the lack of over-the-gap excitations, while still permitting sub-gap transport, implies the existence of gapless edge states.
3. $s_{z}$ conservation is necessary for decoupling spin up and spin down species.

I hope that helped.