I understand how to measure the energy of electron in experiment by ARPES. Could you please explain how to get the picture of energy as function of momentum?
Good question. You can calculate the density of states from the imaginary part of the retarded Greens functions in real space on the lattice. Then you can Fourier transform this to momentum space.
In case of infinite 1d chain, due to symmetry [Gºa]11 = [ Gºb]11 = greens function corresponding to the uncoupled with the other semi infinite part but coupled to its own semi infinite part then won't the greens function for [Gºa] = z/2t² - i/t sqrt(1- (z/2t)²) . Why did you subsitute it with z which would have been the case if site A would not have been coupled to the semi infinite chain A. Can you please clarify where I am going wrong? 1:09:39
Hey buddy, did you find the answer? Then plz reply me. But I approached a convention, [G°a]^-1 is the uncoupled green's function for the system a, as it is uncoupled with the b part but coupled with the semi-infinite a part. So according to you, it just can't be written a simply "z". But it can be further expanded into two parts, for the very first element of system a, say a1 and the other semi infinite part a2. Now [G°a]^-1 =[ G°a1]^-1 - t²[G°a2]. And then you can see, the 2nd part added to the t²[G°b] part, making it 2×t²[G°b] and [G°a1]^-1 remains as z.
Great! I really like your style of teaching! Is it possible to get a finite width spectral function in non-interacting system? My professor presented an example of a 1D chain with a chance that a site has an impurity (other onsite energy). He ended up with a Green's function where the thing which you called "hybridization energy" (we called it self-energy) has actually an imaginary part which of course leads to a finite lifetime. During the calculation he did some thing which were not clear (even after asking him)..
Thanks for the question! Yes, non-interacting systems can certainly have a finite bandwidth. A simple example would be the infinite or semi-infinite homogeneous 1d tight binding chains discussed in these lectures. With a constant tunneling matrix element t between neighbouring sites, the bandwidth is 4t. For systems with an infinite number of sites (the thermodynamic limit) these bands can have a continuous density of states. I like to reserve the term 'self-energy' for the correction to the Green's function due to interactions (while I call the correction due to coupling to a non-interacting bath a 'hybridization'). However, some people use self-energy to mean either/both. You will see either convention in books and research articles. As long as you define everything and are consistent, it doesn't really matter what you call it, I guess!
@@drmitchellsphysicschannel2955 Thanks for your answer! I formulated my question in the wrong way and need to be more specific. Of course, in the continuum limit (infinitely many sites), a 1D chain has a continuum of states, the band. However, my professor introduced some kind of disorder (impurity sites) to the system, through which the Green's function get an additional term in the denominator (other than the bandstructure e_k). This term had a FINITE imaginary part, which would lead to an exponential decay of a k-state, i.e. a finite lifetime. Tke k-states from the periodic bare system somehow "scatter at the impurities". I thought this was only possible for interacting electron systems.
One of the best lectures I've heard. Crisp and clear!!!
Kudos to you Professor!
I understand how to measure the energy of electron in experiment by ARPES. Could you please explain how to get the picture of energy as function of momentum?
Good question. You can calculate the density of states from the imaginary part of the retarded Greens functions in real space on the lattice. Then you can Fourier transform this to momentum space.
Thank you very much. Your videos are very helpful.@@drmitchellsphysicschannel2955
Hi, is there similar Dyson equation for the lesser Green's function?
Hi, @Dr Mitchell's physics channel, where I find the derivation of the theorem about matrix in 1:00:37 ?
Here is the original reference I found for this:
doi.org/10.1016/S0898-1221(01)00278-4
In case of infinite 1d chain, due to symmetry [Gºa]11 = [ Gºb]11 = greens function corresponding to the uncoupled with the other semi infinite part but coupled to its own semi infinite part then won't the greens function for [Gºa] = z/2t² - i/t sqrt(1- (z/2t)²) . Why did you subsitute it with z which would have been the case if site A would not have been coupled to the semi infinite chain A. Can you please clarify where I am going wrong? 1:09:39
Hey buddy, did you find the answer? Then plz reply me.
But I approached a convention, [G°a]^-1 is the uncoupled green's function for the system a, as it is uncoupled with the b part but coupled with the semi-infinite a part. So according to you, it just can't be written a simply "z". But it can be further expanded into two parts, for the very first element of system a, say a1 and the other semi infinite part a2. Now [G°a]^-1 =[ G°a1]^-1 - t²[G°a2]. And then you can see, the 2nd part added to the t²[G°b] part, making it 2×t²[G°b] and [G°a1]^-1 remains as z.
Great! I really like your style of teaching!
Is it possible to get a finite width spectral function in non-interacting system?
My professor presented an example of a 1D chain with a chance that a site has an impurity (other onsite energy). He ended up with a Green's function where the thing which you called "hybridization energy" (we called it self-energy) has actually an imaginary part which of course leads to a finite lifetime. During the calculation he did some thing which were not clear (even after asking him)..
Thanks for the question! Yes, non-interacting systems can certainly have a finite bandwidth. A simple example would be the infinite or semi-infinite homogeneous 1d tight binding chains discussed in these lectures. With a constant tunneling matrix element t between neighbouring sites, the bandwidth is 4t. For systems with an infinite number of sites (the thermodynamic limit) these bands can have a continuous density of states. I like to reserve the term 'self-energy' for the correction to the Green's function due to interactions (while I call the correction due to coupling to a non-interacting bath a 'hybridization'). However, some people use self-energy to mean either/both. You will see either convention in books and research articles. As long as you define everything and are consistent, it doesn't really matter what you call it, I guess!
@@drmitchellsphysicschannel2955 Thanks for your answer! I formulated my question in the wrong way and need to be more specific. Of course, in the continuum limit (infinitely many sites), a 1D chain has a continuum of states, the band. However, my professor introduced some kind of disorder (impurity sites) to the system, through which the Green's function get an additional term in the denominator (other than the bandstructure e_k). This term had a FINITE imaginary part, which would lead to an exponential decay of a k-state, i.e. a finite lifetime. Tke k-states from the periodic bare system somehow "scatter at the impurities". I thought this was only possible for interacting electron systems.