How can I compute edge states in the Haldane model with varying dimensions?

[Ammo1337]

Hi there, studying the Haldane model for my final year project next year, having some trouble understanding how the distant and nearest neighbour hopping works with varying width or height of a 2d sheet in the Haldane model? I'm find some references in papers but they appear to be way over my head and not accessible at all as I've only touched solid state physics (tight binding model and such) and not much condensed matter.

I'm also very interested to know how these hopping terms (NN or distant) are accounted for when looking at edge states in a bulk model?

I've also looked at how these edge states are calculated but I'm not having much luck. I assume it's something obvious and just the resulting energy spectrum from band theory and the tight binding model?

I would appreciate any kind of push in the correct direction, I've asked a lot of questions here but I'm a little scatter brained at the moment! I'll be doing a lot of reading over the weekend

 

[SpinFlop]

Previous thread with link to fantastic article:

https://www.physicsforums.com/threads/is-there-an-easy-review-about-topological-insulator.845717/

 

[king vitamin]

I'm used to the Haldane model being defined with just nearest and next-nearest neighbor couplings. When you say "distant," are you referring to a variation of Haldane's model?

I've also looked at how these edge states are calculated but I'm not having much luck. I assume it's something obvious and just the resulting energy spectrum from band theory and the tight binding model?

What I think should work is to calculate the spectrum on an infinite strip or a cylinder. You should find zero-energy states in the topological phase, and argue that they live on the edges.

 

[Ammo1337]

I'm used to the Haldane model being defined with just nearest and next-nearest neighbor couplings. When you say "distant," are you referring to a variation of Haldane's model?
What I think should work is to calculate the spectrum on an infinite strip or a cylinder. You should find zero-energy states in the topological phase, and argue that they live on the edges.

Yes I've looked at this and you are correct, I've managed to get this correct and solve the bulk, however I am having trouble creating a general hamiltonian that can represent a graphene lattice with both NN and NNN hoppings that is n lattice sides wide in the x direction and m in the y direction, so I am able to analyse edge currents for varying sizes, eg 10x10, 5x5, 100x100, as the haldane model.

After this I will be looking into armchair and zigzag edge states respectively, and determining their boundary conditions

I'm having quite a bit of trouble with this, are there any lecture notes out there of this? Papers don't seem to be helping much as they usually skip all this and just include the graphs and numerical results.

Many thanks for any help you can provide once again!

Previous thread with link to fantastic article:

https://www.physicsforums.com/threads/is-there-an-easy-review-about-topological-insulator.845717/

This has helped quite a bit with starting me off, thank you.

 

[king vitamin]

There is a textbook, "Topological Insulators and Topological Superconductors" by Bernevig and Hughes, which computes the edge states in graphene. This should be a good reference for how to do this for the Haldane model, which is defined on the same lattice. Unfortunately, the book is pretty awful (it's riddled with typos and mistakes), but I can' think of another resource with the specific computation you're requesting.