Coupling to an electric field in a tight binding model

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Discussion Overview

The discussion revolves around the coupling of electrons to an electromagnetic (EM) field within the framework of a second quantized tight binding model. Participants seek references and explanations regarding the Peierls substitution and its implications for calculating current in this context.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant describes the Peierls substitution, which involves modifying the hopping parameter by an exponent of the line integral of the vector potential along the hopping path, and expresses a need for further understanding and references.
  • Another participant provides links to articles that may serve as references for the discussed formalism.
  • A request is made for a good derivation of the Peierls substitution, specifically referencing equations from a paper.
  • Participants discuss whether certain equations from a specific paper are sufficient for deriving the Peierls substitution.
  • There is a mention of the generality of an equation relating the momentum and position functions, indicating a deeper mathematical relationship.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the derivation and implications of the Peierls substitution. There is no consensus on the sufficiency of certain references or equations, and the discussion remains unresolved regarding the best sources for derivation.

Contextual Notes

Limitations include the participants' varying familiarity with the mathematical formalism and the original works related to the Peierls substitution, which may affect their ability to fully grasp the concepts discussed.

Qturtle
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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!
 
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Qturtle said:
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
 
Last edited:
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 ):
 
Don't equation 9 and 10 of Vogl and Graf suffice as a derivation?
 
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)
 
Equation 9 is an exact relation for any function of p and x. Just use ##p=\frac{\hbar}{i} \partial_\vec{r}##.
 
yes I can see now that this is more general. Thanks
 

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