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 of retarded
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 :
G^{r} (E) = -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”).
Thanks a lot!
On Sunday, September 15, 2019, Abbout Adel <abbout.a...@gmail.com> 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 <quantum....@gmail.com>
> 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
>
