# Impurity State Green's functions

1. Aug 26, 2011

### maverick280857

Hi

I am trying to understand the role of Green's functions in the analysis of a 1D chain of on-site impurities (with the impurity state potential amplitude proportional to V). This is in the context of a course on condensed matter field theory that I am trying to follow.

Can someone explain me the following equation

$$G_{ij} = G_{ij}^{0}(\omega) + G_{i0}^{0}(\omega)\frac{V}{1-VG_{00}^{0}(\omega)}G_{0j}(\omega)$$

Here $G_{ij}^{0}(\omega)$ is a 'free' two point Green's function. Here the indices refer to 'sites' (the impurity potential is an on-site potential).

What is the origin of this equation? How does the $1-VG_{00}^{0}(\omega)$ term appear in the denominator?

EDIT -- I understand this is a kind of Dyson equation...trying to figure out the notation though. Appreciate inputs.

2. Aug 28, 2011

### PRB147

tell me the book, so I can follow some omission

3. Aug 28, 2011

### shawl

The intermediate term is the Self-energy term. It is calculated by infinite iteration. It also can be solved in the limit of Born approximation
The detail solution you may find please refer to the "Many particle physics" authored by G. Mahan.
Please take a good book to learn Green's function, or you will be confused and further lose your interests.

4. Aug 29, 2011

### stone

Could you please suggest some good books. I tried Mahan but it was a bit too difficult for me. Is there something at a more beginner's level?

5. Sep 10, 2011

### maverick280857

Okay, it seems I have since figured out some things :-)

A good book is Fundamentals of Many Body Physics by Nolting. If it is too advanced, consider looking at the books by Doniach and Sondheimer, or Fetter and Walecka. My question was not on Green's functions but rather on Impurity Sites. I had the wrong picture for it. Managed to figure it out myself..thanks to those who replied.