Green function for the particle hopping on a lattice, meaning?

[bulgakov]

I am having trouble understanding something that I am sure is very basic. Let's say I have a particle that is hopping on a 1d lattice with a hard wall at x=0 in the presence of some potential - anything, say linear ##H_0=F*i## or Coulomb ##H_0=C/i## where i is the label of the site the particle is on. I treat nearest neighbor hopping as a perturbation. I then calculate the self energy for ##i=1## by adding up all the diagrams that correspond to the particle hopping away from site 1 and then back to it. Using that self-energy I obtain the expression for the Green's function ##G_1(w)## (w is the complex frequency/energy, 1 stands for site 1) and the imaginary part of it gives me the density of states.

Now my question: ahem, what did I just calculate, exactly? For an attractive Coulomb potential this gave me an infinite number of delta functions - bound states, followed by a continuum of scattered states which look right for the Coulomb potential problem on a lattice. Why is it that I get all the eigenvalues of the system from ##Im[G_1]##? Naively, I expected to get ##E_1## from ##G_1##, ##E_2## from ##G_2##, etc. Would I get the same eigenvalues if I calculated ##G_2##? (I know I could check by actually doing it, but either way, I clearly don't understand what I am doing even though I seem to be getting correct results).

Any help would be appreciated :)

 

[eljose79]

Thank you for sharing your question with us. It seems like you are on the right track, but may be missing some key concepts in your understanding of what you have calculated.

Firstly, by calculating the self-energy for site 1, you are essentially taking into account the interactions between the particle at site 1 and all other sites in the lattice. This is why you are getting an infinite number of delta functions in the case of an attractive Coulomb potential - you are accounting for all possible bound states that can exist in the system.

Secondly, the Green's function is a mathematical tool that allows us to calculate the probability amplitude for a particle to propagate from one point to another in a system. In your case, by calculating the Green's function for site 1, you are essentially calculating the probability for the particle to hop away from site 1 and then back to it. This is why you are getting the bound states and continuum of scattered states in the imaginary part of the Green's function.

To answer your question about why you are getting all the eigenvalues from the imaginary part of the Green's function, it is because the eigenvalues correspond to the poles in the Green's function. These poles represent the energies at which the system can sustain a bound state. In your case, you are getting the bound state energies by calculating the imaginary part of the Green's function.

Finally, to answer your question about whether you would get the same eigenvalues if you calculated the Green's function for site 2, the answer is no. This is because the Green's function is site-dependent and the bound states will depend on the potential at each individual site.

I hope this helps clarify your understanding of what you have calculated. It is always important to have a solid understanding of the concepts behind the calculations, so don't hesitate to reach out for further clarification if needed.