Suppose we have a momentum space tight binding Hamiltonian $H(\vec{k})$ that describes some topologically ordered system. It could be a Chern insulator in two dimensions, or a Weyl semimetal in three dimensions - basically, anything that supports edge states of some kind.

What I'm trying to do is understand the procedure for finding the dispersion relation for these edge states. It seems to me that we want to find a way to cleave our infinite system along some surface, and somehow use our knowledge of the bulk, infinite system Hamiltonian to project the band structure onto that surface. However as a set of mathematical operations it's not really clear to me how to go through with this procedure, starting with the infinite system tight binding model.

The physics of what's going on here is explained nicely in some review articles that I've read, but I haven't found any tutorials actually explaining how to extract these edge modes in detail. Any help getting pointed in the right direction would be very welcome.

Let's assume that we can compute the bulk band structure in full detail and proceed from there.

  • $\begingroup$ Numerically or analytically? Also note that gapped and gapless systems might require rather different methods, particularly for analytical methods. $\endgroup$ – Anyon Mar 19 at 0:35
  • $\begingroup$ I'm thinking numerically right now. I assume that the techniques in those cases will be broadly similar, but the way you actually set up the calculations will depend on the system in particular. I'm more interested in semimetals but I think a description in either case will be useful. $\endgroup$ – miggle Mar 19 at 1:06

The short answer is that if you want to find edge states, you'd better introduce an edge. In other words, it's not enough to solve the bulk band structure assuming a nice periodic lattice for which we can use a nice momentum space basis.

You can find a clear analytical treatment of edge physics in quantum Hall and topological insulator states in Fradkin's Field theories of condensed matter physics (Cambridge), 2nd edition, chapters 15 and 16. An early argument for Fermi arc surface states in Weyl semimetals was laid out by Wan et al. (2010) by building on the quantum Hall description, so it is a good starting point.

Anyway, the basic idea is to use open boundary conditions in at least one dimension, and possibly introducing a confining potential to keep the particles in the system, and then working out the implications for states near the Fermi level. This is rather straightforward in the integer quantum Hall case, where the confining potential leads to bending of Landau levels. (Another reason for starting from there.)

To numerically demonstrate the presence of edge states the idea is the same* - you introduce an edge and deal with a system with at least one finite dimension. For a Chern insulator or topological insulator, typically a stripe geometry is used - which just means infinite along $x$, finite along $y$. You just need to make the finite dimension large enough that you can still distinguish bulk from edge physics, but still manageable by whatever solver you happen to be using. (I'm not sure what the typical geometry for Weyl semimetals would be.) For Weyl semimetals and Chern insulators, you might just want to start with a tight-binding model that you diagonalize directly. Beyond that, how you set up the calculation starts to depend on the system in question, and the band structure method used. (E.g. in DFT, I believe you'd need to use a supercell to achieve open boundary conditions.)

  • $\begingroup$ Thanks a lot for your answer! I understand the general idea here, but I'm confused about how one actually sees the edge modes. If I have some finite number of bulk bands $N$, and I follow the prescription you just described (i.e., repeat the calculation with one or more sets of open boundary conditions), then I should still have $N$ bands that I imagine will look a lot like the bulk, but with some finite size corrections. Is there some additional step that you need to carry out to actually see states on the physical edge of the system? $\endgroup$ – miggle Mar 21 at 3:34
  • $\begingroup$ @miggle Well, if we consider a topological insulator, you should see well-separated bulk bands and in-gap states corresponding to bulk states. To directly show that they're at the edge you'll need to look at some kind of local quantity - be it a local (or surface) density of states, local spectral weight, a current density, a local Chern marker etc. If you obtain the band structure in momentum space, you'll usually need to go to an intermediate real-space representation (e.g. Wannier functions) to calculate such quantities. $\endgroup$ – Anyon Mar 21 at 14:28

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