Interpreting the Greens function in frequency space

[aaaa202]

I have done a problem with a spin two electron spin system. I have found the Greens function propagator for spin up->spin up, which I have written in a form where the occupancy of spin up electrons is incorparated (attached) and from there I can get the spectral function, which will then be something like (omitting some factors):
A=(1-N_up)δ(ω-E-U) + N_upδ(ω-E)
But I am not sure how to interpret it. I know that for free electrons the spectral function A(k,ω) is something about how we can add an electron with momentum k to the system only by adding an electron with energy ω=E. But this can't be translated directly to how we can add an electron with spin up to the system since that would be impossible for 2 of the possible 4 configurations due to pauli exclusion. I can see that if the there is one electron of spin up the spectral function yields ω=E while if there is 0 ω=E+U. But how exactly is it to be interpreted?

 

[mark726]

Thank you for sharing your findings on the spin two electron spin system. It seems like you have made some interesting progress in understanding the system and its properties.

Firstly, let's discuss your question about interpreting the spectral function. As you mentioned, the spectral function for free electrons tells us about the possibility of adding an electron with a specific momentum and energy to the system. However, in your case, the spin of the electron also plays a role in determining the possible configurations. Therefore, the spectral function for the spin two electron spin system can be interpreted as the possibility of adding an electron with a specific spin, momentum, and energy to the system.

In your equation, the first term represents the possibility of adding an electron with spin up to the system, where the energy of the electron is equal to the energy of the system (ω=E). This term is multiplied by (1-N_up), which accounts for the fact that there is already an electron with spin up in the system, and therefore the probability of adding another electron with spin up is reduced. The second term represents the possibility of adding an electron with spin up to the system, where the energy of the electron is equal to the energy of the system plus the energy gap U (ω=E+U). This term is multiplied by N_up, which accounts for the fact that there is no electron with spin up in the system, and therefore the probability of adding an electron with spin up is increased.

Overall, your spectral function provides information about the possible configurations of the system, taking into account the spin of the electrons. I hope this helps to clarify the interpretation of your results.