Green's functions for interacting fermions

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Dr Mitchell's physics channel

Күн бұрын

Пікірлер: 14

@MinecraftworksRUS Жыл бұрын

I found the part of this lecture series about Greens functions very helpful. Thanks!

@Hamza-vk6sc 4 ай бұрын

Thank you for the lecture professor, So, both methods (Equation of Motion and Layman’s representation) don’t work with interacting particles. But how do theorists calculate the spectral functions like XAS and XPS for very complicated materials? Is it done using cumulant Green functions or something else? My question might not be well-formed, but I'm completely new to this field. and if there any materials can help a beginner's like me I will appreciate. Thank you for considering my question.

@drmitchellsphysicschannel2955 4 ай бұрын

Good question! In principle, equations of motion and Lehmann representation are generally applicable and can be applied to interacting systems. In practice, straight application of equations of motion does not work because you just generate an exponentially-large system of equations to solve, so it does not get you anywhere. However, EoMs are used to produce approximation schemes for interacting Green's functions in theoretical physics research (for example, the so-called "hierarchical equations of motion" method, HEOM). Lehmann is however very general and is widely used for interacting Green's functions -- but this is all done numerically. Exact diagonalization and Tensor network methods for example would compute the matrix elements of creation and annihilation operators directly and do the Lehmann sum to get a discrete representation of the spectral function. Depending on the application, the poles of the spectrum might then be broadened to form a continuous representation of the "true" spectrum. I did not cover it in these lectures, but there are other ways to compute the Green's function which might be used in materials science, for example computing imaginary-time / Matsubara Green's functions using continuous time Monte Carlo methods.

@liukelyu6009 3 жыл бұрын

At 21:16, when T go to infinity, shouldn't Fermi Dirac distribution go to 1/2?

@drmitchellsphysicschannel2955 3 жыл бұрын

Thanks for this correction! Yes, the Fermi function goes to 1/2 as T->infinity. So the occupation goes to 1/2 and both poles are of equal weight in the spectral function.