The reason for discrepancy between the scattering wave function and Greens
function is extremely subtle. I have now figured out the reason and it is
most definitely not due to missing velocity like you suggested.
Essentially, Psi Psi^\dag boils down to the imaginary part of the Greens
function for *incoming mode from the left lead only*.
Thanks for your input. I appreciate it.
On Sun, Sep 15, 2019 at 4:53 PM Amrit Poudel <quantum....@gmail.com> 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 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
>>
>
