A How to get Peierls substitution in edge state?

[haw]

In paper PRL 101, 246807 (2008), authors use "Peierls substitution", that is ky -> -i∂ y. As we know, ky is eigenvalue of translation operator in period potential, while -i∂ y is momentum operator, it seems they are huge different. So I wonder how to get ""Peierls substitution" in strict math way?

 

[king vitamin]

In paper PRL 101, 246807 (2008), authors use "Peierls substitution", that is ky -> -i∂ y. As we know, ky is eigenvalue of translation operator in period potential, while -i∂ y is momentum operator, it seems they are huge different. So I wonder how to get ""Peierls substitution" in strict math way?

No, k_y is the eigenvalue of momentum p_y = -i \partial_y.

The translation operator in the y direction is given by T_y(a) = e^{-i p_y a}.

 

[DeathbyGreen]

Peierls substitution is a way to couple a tight binding Hamiltonian to a external magnetic field within the lattice approximation. I see what you are referring to in the paper; they say that they use that substitution to say k_y \rightarrow \partial_y. I think they might be misusing the term; When they move from PBC's to finite BC's along the y direction, k_y is no longer a good quantum number. And in moving from a lattice model with momentum k_y to a continuum model with crystal momentum \hbar k they make the substitution k_y \rightarrow \hbar k_y or \partial_y. I'm not sure why they call it a Peierls substitution, it looks more like a substitution like lattice to continuum model. Here is a nice forum post about the math behind the Peierls substitution: https://physics.stackexchange.com/questions/178003/tight-binding-model-in-a-magnetic-field

I've actually solved this model before, and the way to do it is to take the momentum space Hamiltonian and do a partial Fourier transform along the y direction, so that in the final product you have a Hamiltonian that is PBC in the x direction but lattice model in the y direction.

 

[haw]

Thanks for your help! Actually helpful.