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.