Dear Amrit, The reason I said your deduction of the Greenfunction formula should be wrong is simple: if for example you have the simple case of only one site as a scatterer, you formula " i psi psi^\dagger" will give you a greens function pure imaginary, which is of course incorrect since you can get the exact result analytically and check that.
The procedure I was mentioning, is the standard method that you can find in the book of Economou for example, where the integration from k to E will involve the density dk/dE (which is the inverse of a velocity.) Now. If your aim is to calculate the exact Greens function with kwant, the method is as follows: Kwant calculates the greens function solely between the sites connected to leads kwant.greensfunction(syst, out_leads=[0], in_leads=[0]) will give you the subblock of greens function between the sites connected to lead "0" If you want the greens function at a given site, you can do that as suggested by Viacheslav Ostrouk [1], by mounting a virtual lead with a self energy 0 to your site. If this virtual lead is number 3 than g= kwant.greensfunction(syst, out_leads=[3], in_leads=[3]). If your intention is to check the formula 25 in the reference you shared, that probably needs more thinking and reading of the details in the reference. I hope this helps Adel [1] https://mailman-mail5.webfaction.com/pipermail/kwant-discuss/2016-January/000720.html On Mon, Sep 16, 2019 at 9:51 AM Amrit Poudel <[email protected]> wrote: > Hi Adel, > > Thanks a lot for your quick response. I appreciate it. > > Please note that I have carefully tested the two approaches (see below) in > a clean 1D system without random onsite potential. Only in that case, two > approaches match, so my equations are correct. However, when on-site > potential is present, then two approaches do not match at all. > > I just want to clarify a few things: > First, in 1D, there is only one mode present, so both wave function and > Greens function approaches take into account the same number of modes. > There are no additional modes that Greens function somehow finds and takes > into account. This cannot be the reason for discrepancy. > > Second clarification is, my approach (I) is direct inversion: > G^{r}(E) = [E+i\eta - H - \Sigma^r_{leads}]^{-1} . > I can directly invert a small matrix to compute G^{r}. I don’t use this > formula to derive my wave function approach. This is used to test whether > Greens function computed using scattering wave function is accurate or not. > > My approach (II) is based on *scattering wave function* (note the emphasis > on scattering, these are not eigen modes!!) of the scattering region of the > system. The connection between the scattering wave function and retarded > Green’s function is provided in Eq. 25 of the reference I quoted earlier. > Please check the reference before you do your derivation, which I believe > is incorrect. You are using equations that is neither related to Eq 25, nor > related to what I am using. > > In that reference, Green’s function is given in time domain (t, t’). It is > a straightforward task to write that in frequency or energy domain since > the system is time independent, so G is time translationally invariant > (t-t’). The energy domain expression is: > > G^r(E) = -i \Psi_{\alpha}(E)*\Psi^\dag_{\alpha}(E). > > Here \Psi_{\alpha}(E) is a *scattering wave function* at arbitrary energy > E. Again, I want to emphasize that these are scattering states not some > eigenmodes. > > I would appreciate if you could start your own derivation for Greens > function using *scattering wave function* (please not that our system is > open system) starting from Eq. 25 of the reference I mentioned. You will > end up with > -i \Psi_{\alpha}(E)*\Psi^\dag_{\alpha}(E) for a stationary or time > independent system. > > If you run my Python example file, which I included in my previous email, > by removing the on-site potential (comment out that line), you will see > that two approaches exactly match and both give the same Greens function. > If I had missed a velocity or something like you said, then two quantities > would not have matched in a clean 1D system. Please try my code yourself > and see for yourself (a small typo in the code in the if statement for > onsite potential, instead of “or” use “and”). > > > I would really appreciate if we could resolve this discrepancy. I look > forward to any suggestions. > > Thanks for your time! > > On Sun, Sep 15, 2019 at 7:27 AM Abbout Adel <[email protected]> wrote: > >> Dear Amrit, >> >> You wrote: >> " this is based on Eq. 25 in energy domain of >> https://arxiv.org/pdf/1307.6419.pdf " >> and deduced that G^r(E) = -i*\Psi(E)*\Psi^\dag(E). >> >> Your deduction must be wrong. I suggest to you to do the calculation for >> a simple case and test your formula in a clean 1D system. >> G^r=1/(EE-H+i eta) ==> G^r_ij= Sum_k Psi_ik 1/(EF-E_k+i eta) Psi_kj >> >> The sum over k can be changed to a an integral over energy >> G^r_ij= Integral Psi_i(E) 1/(EF-E_k+i eta) Psi_j(E)^dagger * dk/dE >> *dE >> >> when you carry on the calculation,you see that there is a term which is >> missing in your formula which is dk/dE. This term is the inverse of the >> velocity. If you carry the calculation for the 1D case, you will find the >> correct form known in literature. >> >> I want to bring to your attention that for Greens energy calculation, the >> evanescent modes are also taken into account and not only the propagating >> ones (the one obtained by kwant. wavefuntion are propagating) . >> >> I hope this helps, >> Regards, >> Adel >> >> On Sat, Sep 14, 2019 at 9:59 PM Amrit Poudel <[email protected]> >> wrote: >> >>> Hello Kwant users, >>> >>> I am trying to compare the retarded Green's function of a simple 1D wire >>> attached to two leads using two different methods: >>> >>> (I) Using scattering wave function obtained from the Kwant software >>> and using Eq. 25 in energy domain of "Numerical simulations of time >>> resolved quantum electronics (https://arxiv.org/abs/1307.6419) written >>> by Kwant authors. >>> >>> (II) Direct computation of the retarded Green's function by inverting >>> device Hamiltonian with self energies of the attached leads (again computed >>> from the Kwant software): G^r(E) = [E+\i*\eta - H- \Sigma^r_{leads}]^{-1} >>> for a relatively small system size (5 sites in the attached example). Here >>> both H and \Sigma^r are computed from the Kwant software. >>> >>> However, I find that these two results do not agree even in a simple 1D >>> example when on-site potential is present in the few sites of device or >>> scattering region only. >>> >>> I have attached the Python script with this email. >>> >>> Does anyone know the reason behind the discrepancy between the two >>> methods?I would greatly appreciate any comments/suggestions on how we can >>> resolve this error? >>> >>> Thanks! >>> >>> >>> >>> >> >> -- >> Abbout Adel >> > -- Abbout Adel
