Impurity State Green's functions

In summary, the equation is a generalization of the Fokker-Planck equation. It describes the interaction between particles. 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.
  • #1
maverick280857
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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

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

Here [itex]G_{ij}^{0}(\omega)[/itex] 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 [itex]1-VG_{00}^{0}(\omega)[/itex] term appear in the denominator?

EDIT -- I understand this is a kind of Dyson equation...trying to figure out the notation though. Appreciate inputs.
 
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  • #2
tell me the book, so I can follow some omission
 
  • #3
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
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
stone said:
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?

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.
 

1. What is an impurity state Green's function?

An impurity state Green's function is a mathematical tool used in quantum mechanics to describe the behavior of an electron that is localized to a specific site, such as an impurity in a crystal lattice. It takes into account the interactions between the localized electron and the surrounding electrons in the system.

2. How is an impurity state Green's function different from a regular Green's function?

An impurity state Green's function differs from a regular Green's function in that it includes the effects of localized electrons, while a regular Green's function does not. It also has an additional term called a self-energy, which describes the interactions between the localized electron and the surrounding electrons.

3. What is the significance of impurity state Green's functions in condensed matter physics?

Impurity state Green's functions are important in condensed matter physics because they allow us to study the behavior of localized electrons, which are common in materials with impurities or defects. This is essential for understanding the electronic properties of materials and developing new technologies.

4. How are impurity state Green's functions calculated?

Impurity state Green's functions are typically calculated using advanced mathematical techniques, such as perturbation theory or numerical methods. These calculations take into account the interactions between the localized electron and the surrounding electrons, as well as any external perturbations or disorder in the system.

5. What are some real-world applications of impurity state Green's functions?

Impurity state Green's functions have a wide range of applications in condensed matter physics, including the study of electronic properties in materials, the development of new electronic devices, and the investigation of superconductivity and magnetism. They are also used in fields such as quantum computing and nanotechnology.

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