Fermi Surface and Orthogonality Catastrophe

In summary, the author is discussing the discontinuous jump in the occupation number of a system, which is of size Z. This value, Z, is equal to the probability that the system will be in an N+1 (N-1) ground state after an electron is added (removed). The author claims this can be derived from equation 8.123, but it is not clear how. The above relation is based on the following equation: Z = |\langle 0,N+1|a^\dagger |0,N\rangle|^2 = |\langle 0,N-1|a |0,N\rangle|^2.
  • #1
Morberticus
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I am reading section 8.5.1 of http://f3.tiera.ru/2/P_Physics/PS_Solid%20state/Giuliani%20G.,%20Vignale%20G.%20Quantum%20theory%20of%20the%20electron%20liquid%20%28CUP,%202005%29%28ISBN%200521821126%29%28799s%29_PS_.pdf (page 442 of the book, page 465 of the pdf). The author claims the discontinuous jump in the occupation number is of size Z, where

[itex]Z = |\langle 0,N+1|a^\dagger |0,N\rangle|^2 = |\langle 0,N-1|a |0,N\rangle|^2[/itex]

I.e. Z is the probability that the system will be in an N+1 (N-1) ground state after an electron is added (removed). They say this follows from equation 8.123 but I don't find it obvious how. Is there an easy logical path from equation 8.123 to the above relation?
 
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  • #2
I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?
 
  • #3
post equation 8.123 for us
 

1. What is a Fermi surface?

A Fermi surface is a concept in solid state physics that describes the boundary between occupied and unoccupied energy levels of electrons in a material at absolute zero temperature. It represents the points in momentum space where the energy of the electrons is equal to the Fermi energy.

2. What is the significance of Fermi surfaces?

Fermi surfaces provide important information about the electronic properties of a material, such as its electrical conductivity, thermal conductivity, and magnetic properties. They also play a crucial role in determining the behavior of electrons in a material, including their interactions with each other and with external fields.

3. What is the orthogonality catastrophe in relation to Fermi surfaces?

The orthogonality catastrophe is a phenomenon that occurs when the wave functions of electrons in a material are abruptly changed due to a perturbation, such as an applied electric field. This results in a breakdown of the orthogonality between the initial and final states of the electrons, leading to a rapid change in the energy of the electrons and the formation of new energy levels.

4. How is the orthogonality catastrophe related to the Fermi surface?

The orthogonality catastrophe is closely related to the Fermi surface because it is a result of the strong interactions between electrons near the Fermi surface. These interactions cause the electrons to rearrange their positions and energies in response to a perturbation, leading to the orthogonality catastrophe.

5. What are some potential applications of studying Fermi surfaces and the orthogonality catastrophe?

Understanding Fermi surfaces and the orthogonality catastrophe can provide insights into the properties and behaviors of materials, which can have practical applications in fields such as materials science, condensed matter physics, and electronics. It can also aid in the development of new materials with desired electronic properties.

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