[Kwant] Re: Inquiry of applying Bound State Algorithm on 3D Systems

Dear Ryan and Chi, To complement Mathieu’s email: * Indeed, it might be a good idea for you to look at https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.1.03318... https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.1.03318... which might solve your problem * Keep in mind that in a 3D TI, the bound state are bound only along one direction (say z) and delocalized along the others (x and y). * Both approaches are possible: take a system large but finite along x and y and infinite along z — OR — work in (kx,ky) with a fully infinite system and perform a fourrier transform back to (x,y) at the very end. The second one allows one to work directly in the thermodynamic limit and has been used in the PRR paper above. I think it is better but the first one might suffice, depending on what you want to do (however you will have to check for finite size effect due to finite length in x and y direction). Best regards, Xavier

Le 18 août 2021 à 11:14, Tiew, Hoe R a écrit :

Good morning,

We are writing this email as we would like to ask about the application of the bound state algorithm (described in https://scipost.org/10.21468/SciPostPhys.4.5.026 https://scipost.org/10.21468/SciPostPhys.4.5.026) to higher-dimensional systems. In particular, we are trying to apply the algorithm to solve for proximitised 3DTIs, and for reference have been trying to understand the example cases of the quantum billiards and 2D topological insulator in the paper. Essentially, we would like to clarify if the reduction of higher-dimensional systems to quasi-1D is necessary for the bound state algorithm to work correctly.

To solve for the 2D BHZ model (Section 5.2), the system was reduced to a quasi-1D system through the use of the Bloch theorem, and was then fed into the bound state algorithm. In section 4 however, a system of quantum billiards was considered without any mention of reduction to a quasi-1D system. As we are trying to apply the algorithm to a 3D system, we are wondering if this is a necessary step for the algorithm to work as intended. In the proximitised 3DTI systems we are considering, the leads are translationally invariant only in 1 direction, so we are wondering if the algorithm can be applied if we do need to reduce the system like in section 5.2.

Thank you very much!

Regards, Ryan and Chi