Topological Insulator : Edge states

[hbaromega]

I had been reading several articles on topological insulators (TI) including the Kane and Hasan's 2010 RMP. I am not very much clear about the Z_2 invariant TI. I mean, the even-odd argument proposed by Kane and Male (also argued by S. C. Zhang's group and Joel Moore's group in a different way). My question is

How can one get more than two edge states in quantum spin hall effect (QSHE)? Wouldn't it be only two edge states (one for spin up and the other for spin down)?

I'm asking by taking analogy from the integer quantum hall effect (IQHE) which seems to have only one chiral state in either of two edges of the 2D quantum hall system.

 

[ismets]

Hi,
I have just started studying this topic. Can you share your sources because I have a hard time find good resources on topological insulators.

 

[stone]

In IQHE there are two edge states, one at each edge obviously. In this case there is no distinction between up and down spin.

Now in QSH insulators, the spins are also separated. On either edge there are two spin channels. In addition the spins are moving in opposite direction at each edge. So in total there are four channels or edge states as they may like to call.

 

[suntring]

Well, I am also new to this filed.

I was wondering if your confusing point is that since QHE is described by a band theory, then for every energy band you have 2 or 4 edge states. So depending on how many energy levels are filled, let say m, there could be 2m/4m edge states.

Hope this may help.

 

[weejee]

I'm asking by taking analogy from the integer quantum hall effect (IQHE) which seems to have only one chiral state in either of two edges of the 2D quantum hall system.

Integer quantum Hall systems can have any number of chiral edge states, the number of which is equal to the Chern number of the bulk band structure.

How can one get more than two edge states in quantum spin hall effect (QSHE)? Wouldn't it be only two edge states (one for spin up and the other for spin down)?

Any model corresponding to two copies of quantum Hall systems (1) related to each other by time reversal (that means, the Chern numbers associated with the two copies are negative of each other) and (2) with odd Chern numbers, describes a Z_2 non-trivial insulator(QSHE), although this isn't the most general definition of the quantum spin Hall effect.

If the Chern number of one copy is 2n+1, its time-reversed partner should have a Chern number of -(2n+1), and the model will have 2n+1 pairs of counter-propagating edge states (per edge). Still, this number is not robust under disorder, but only the parity is.