Coupling to an electric field in a tight binding model

[Qturtle]

Hi
i'm looking for some references (prefer books) or explanations as to how one couple electrons so an EM field in a second quantized formalism tight binding model.
from what i know, one need to replace the hopping parameter with the same parameter multiplied by an exponent of the line integral of the vector potential along the hopping path. this is called the peierls substitution. after that in order to find the current - one need to calculate the derivative of the Hamiltonian with respect to the vector potential.
Can someone refer me to a book where this formalism is explained, or maybe explain here? i actually need this for a work I'm doing. I know that this is the method to couple to an EM field and finding the resulted current but i don't really understand why.

thank you very much!

 

[DrDu]

Hi
i'm looking for some references (prefer books) or explanations as to how one couple electrons so an EM field in a second quantized formalism tight binding model.
from what i know, one need to replace the hopping parameter with the same parameter multiplied by an exponent of the line integral of the vector potential along the hopping path. this is called the peierls substitution. after that in order to find the current - one need to calculate the derivative of the Hamiltonian with respect to the vector potential.
Can someone refer me to a book where this formalism is explained, or maybe explain here? i actually need this for a work I'm doing. I know that this is the method to couple to an EM field and finding the resulted current but i don't really understand why.

thank you very much!

Not a book, but this article is the reference of the expert:
http://www.wsi.tum.de/Portals/0/Media/Publications/76e71959-883d-474a-b8a0-7569651515fb/PRB51_4940_95.pdf
and this I find also helpful:
http://docs.lib.purdue.edu/cgi/viewcontent.cgi?article=1509&context=nanopub

 

[Qturtle]

Thanks DrDu
i'm actually looking for a good derivation of the peierls substitution (Eq. 9 and rf. 16 in the first paper)
I wasn't able to find the original paper of peierls though ):

 

[DrDu]

Don't equation 9 and 10 of Vogl and Graf suffice as a derivation?

 

[Qturtle]

It is if you understand why an exponent of a line integral of A(x) can transform a function of p to a function of p-eA(x)

 

[DrDu]

Equation 9 is an exact relation for any function of p and x. Just use $p=\frac{\hbar}{i} \partial_\vec{r}$.

 

[Qturtle]

yes I can see now that this is more general. Thanks