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khsrali / LTB-Symm

LTB-Symm is a publicly available code that does two things: large scale tight-binding (LTB) calculation of 2D materials, and checks topological symmetries (Symm) of their wave functions.
https://ltb-symm.readthedocs.io
GNU General Public License v3.0
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Hermitian conjugate #1

Closed LamaKing closed 2 years ago

LamaKing commented 2 years ago

Understand if the H.c. in TB real-space Hamiltonian yields a factor 2 in bandstructure.

khsrali commented 2 years ago

I think that means hermitian conjugate of the whole matrix. not element-wise. To me that is only convenient when one puts just the half of interaction: https://en.wikipedia.org/wiki/%2B_h.c. Otherwise that exactly and only means, a factor of 2. It is very strange to me, why twice transmission integral... I need to understand...

khsrali commented 2 years ago

Yeah, it seems we have to multiply by factor of 2. But please notice, the Hamiltonian must remain complex.

LamaKing commented 2 years ago

Wait, wait! I'm glad you agree, but now I want to write down some example of this to convince myself for good. It's reasonable to me, but I don't want to introduce a factor without proving it must be there. I think the same argument should apply in the phonon-energy of a 1D chain. In that case we can solve it analytically and convince ourselves the factor two is there.

Anyhow, I think the Hamiltonian remains complex, but being Hermitian, the eigenvalues are real. This is what matters: the Hamiltonian is an observable, so its eigenvalues must be real quantities. Makes sense?

khsrali commented 2 years ago

Yes that is true, the Hamiltonian should remain complex, simply because it represents momentom space. Also as we both know, eigen values of a hermitian matrix are always real, independent of complexity of the matrix.

For TB models, H.C always means a factor of 2. Because the first part of Hamiltonian is already hermitian since atoms are always mutually neighbors. To verify I looked at band-gap of mono layer graphene which is often calculated as ~ 11 ev which is 4 t0 and not 2t0. Where t0~2.8

In any case this is just a overall scaling, and has nothing to do with the shape of band structure. One could define 5.6 for the transmission integration instead of that factor of 2. But historicaly it went the other way around, which kinda make sense because physicist could ensure H is hermitian, meaning to add H.C. and to consider t0 half the actual transmission integral ~ 2.8

LamaKing commented 2 years ago

Ok, good argument about the monolayer graphene! Then we can mark this as solved!

khsrali commented 2 years ago

Yes! thanks for pointing this H.C. out at the first place, I was completely ignorant about that..