Unexpected Landau Level Discrepancies in Multilayer Graphene Compared to Published Results
Dear Kwant Group, I attempted to replicate the Landau levels of ABA-trilayer graphene as described in the paper "Landau level spectra and the quantum Hall effect of multilayer graphene," Physical Review B 83, 165443 (2011), Figure 2, using Kwant. The results obtained with Kwant match the paper's results approximately well; however, there is an issue. The number of Landau levels (LLs) under each magnetic field does not match the results presented in the paper. For example (see the attached figure): At a magnetic field strength of ( B = 10 ) Tesla: The left figure shows a total of 46 LLs (18 + 28). The right figure shows a total of 34 LLs (16 + 18). For the specific cases labeled as 'a' to 'd', each of these cases has 2 more LLs compared to a reference. For the case labeled 'e', this case has 4 more LLs compared to a reference. I believe the reference is correct as I was able to reproduce the completely identical LLs results using WannierTools(an open source code which can calculate LLs by Peierls substitution method). I am currently unsure where the problem lies or if it is a bug. I have attached the code (test.py) and a small modification to the spectrum function in plotter.py. Thank you very much to the Kwant development team for your efforts. I hope to get an answer, as it would be very helpful to me. Versions used: 1.4.3 + landau_levels.py Best regards, ac76888 ac76888@sjtu.edu.cn
Dear Yl, I guess this is related to the discretization of the hamiltonian in a square lattice. In tight binding, we often wind up with many issues while descritizing the Dirac Hamiltonian in a square lattice. In the continuum model, it works fine but in tight binding it does not and often winds up with degeneracy that might be related to the doubling fermion. Best Adel Le mar. 18 juin 2024 à 16:51, yl <ac76888@sjtu.edu.cn> a écrit :
Dear Kwant Group,
I attempted to replicate the Landau levels of ABA-trilayer graphene as described in the paper "Landau level spectra and the quantum Hall effect of multilayer graphene," Physical Review B 83, 165443 (2011), Figure 2, using Kwant. The results obtained with Kwant match the paper's results approximately well; however, there is an issue. The number of Landau levels (LLs) under each magnetic field does not match the results presented in the paper. For example (see the attached figure):
At a magnetic field strength of ( B = 10 ) Tesla:
The left figure shows a total of 46 LLs (18 + 28). The right figure shows a total of 34 LLs (16 + 18). For the specific cases labeled as 'a' to 'd', each of these cases has 2 more LLs compared to a reference.
For the case labeled 'e', this case has 4 more LLs compared to a reference.
I believe the reference is correct as I was able to reproduce the completely identical LLs results using WannierTools(an open source code which can calculate LLs by Peierls substitution method).
I am currently unsure where the problem lies or if it is a bug. I have attached the code (test.py) and a small modification to the spectrum function in plotter.py.
Thank you very much to the Kwant development team for your efforts. I hope to get an answer, as it would be very helpful to me.
Versions used: 1.4.3 + landau_levels.py
Best regards,
ac76888
ac76888@sjtu.edu.cn
Thank you for your answer. If we use a honeycomb lattice to discretize the Hamiltonian, can this solve the problem? How should we modify landau_levels.py accordingly? I would appreciate your general suggestions on this matter. Best regards.
Hi, You might test it with Wilson correction. ### ----------------- Wilson correction ------------------------ def onsite_Wilson(site, A): return -(2*A)*sigma_z def hopping_Wilson(site_1, site_2, A): d_projection_xy = site_2.pos - site_1.pos d_12 =d_projection_xy/np.linalg.norm(d_projection_xy) return 1j*A/2*(τ*sigma_x*d_12[0] + sigma_y*d_12[1]) + (A/2)*sigma_z In my case 1j d_12[0] stands for "kx" in *kwant**.**continuum**.* *discretize* and 1j d_12[1] stands for "ky". Ham_wilson = ("A/2 * (k_x * sigma_x + k_y * sigma_y)""+ (-2*A) * sigma_z" You have also to adjust it to your case (A=! depending on your Hamiltonian). See ref1. I don't know what would happen if we used a honeycomb lattice. For sure if you want to test it, we will do it without kwant.continuum.discretize modules. ref1*. Beenakker,et al. Ann. Phys. *(*Berlin*) *2023*, 2300081. *DOI: 10.1002/andp.202300081* I hope this will help. best Le jeu. 20 juin 2024 à 06:45, <ac76888@sjtu.edu.cn> a écrit :
Thank you for your answer. If we use a honeycomb lattice to discretize the Hamiltonian, can this solve the problem? How should we modify landau_levels.py accordingly? I would appreciate your general suggestions on this matter. Best regards.
Dear user, You can try the developpement version of kwant that automatically adds the Peierls substitution on graphene. Automatic addition of Peierls phase terms to Builders What’s new in Kwant 1.5 — Kwant 1.5.0a0.dev103+ge154dc0 documentation (kwant-project.org) <https://kwant-project.org/doc/dev/pre/whatsnew/1.5#automatic-addition-of-pei...> I hope this helps, Regards, On Thu, Jun 20, 2024 at 7:45 AM <ac76888@sjtu.edu.cn> wrote:
Thank you for your answer. If we use a honeycomb lattice to discretize the Hamiltonian, can this solve the problem? How should we modify landau_levels.py accordingly? I would appreciate your general suggestions on this matter. Best regards.
-- Abbout Adel
participants (4)
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Abbout Adel -
ac76888＠sjtu.edu.cn -
Adel Belayadi -
yl