Very good answer Roland! I will take a glance at Fetter & Walecka and at Mahan; do you have any other reference to suggest? Kind regards
Juan Manuel On Sat, Jun 2, 2012 at 12:04 AM, Roland Gillen <[email protected]> wrote: > The poles depicted in the Figure are not those of the retarded Green > function, but those of the complex Fermi-Dirac distribution function n_F. > Obviously, these become important when integrating in the complex plane > instead of the real axis. However, the contribution of the poles to the FD > distribution quickly decreases with pole order (as can be seen in its > Matsubara summation representation), so it's sufficient to only include a > finite number of poles within the integration contour. > As for the shape, well, the integration path in the complex plane is > arbitrary as long as it is closed. So I suppose it was simply chosen in a > way to allow for an efficient handling of the contour integration, enclosing > all the energies larger than EB and a sufficient number of poles of the FD > distribution. > Ozaki apparently chose a different contour for the OpenMX code. > > Roland > > > On 1 June 2012 18:42, Juan Manuel Aguiar <[email protected]> > wrote: >> >> Dear Jonathan, >> According what I know about the NEGF theory, the poles of the Green >> function lie in the real axis. In order to performed the integration, >> the Cauchy's theorem is employed with a semicircular contour centered >> at the origin and slightly displaced of the real axis; then, taking >> the limit of R -> inf, we have the integral computing the poles >> enclosed. >> What I mean by shape is what is depicted in the fig 2 of the paper (I >> enclose it for you to see); I don't understand the reason for such a >> contour. Moreover, the poles of the Green function depicted in the >> figure lie in the imaginary axis resembling the Matsubara frequencies >> for the temperature dependent Green function. Shouldn't be all the >> poles included? the shape chosen for the contour includes only a few >> depending on the value of Delta. >> Let me know if I have to explain better. In a few words, I would like >> to see where is grounded the shape depicted in the figure I enclose, >> the physical meaning and (if it were possible!!) the implications in >> the convergence of the nonequilibrium part in a transiesta the >> calculation. >> Yours >> >> Juan Manuel >> >> On Fri, Jun 1, 2012 at 5:30 PM, Jonthan R. Owens <[email protected]> wrote: >> > Dear Juan, >> > >> > I'm not sure what you mean by shape. Are you asking why it's lifted off >> > the >> > horizontal axis? >> > >> > It is lifted off the horizontal axis to ensure that the Green's function >> > behaves smoothly away from the real axis, allowing for a nice numerical >> > scheme for the integration. You give the distance as a number of poles >> > of >> > the Fermi-Dirac distribution because Cauchy's theorem tells you that the >> > integral of a contour is the sum of the enclosed residues, which are >> > determined by the poles in the integrand. >> > >> > This is pretty much the discussion Brandbyge has in the PRB paper you >> > cite, >> > on pages 4 and 5. >> > >> > If the complex integration is troubling you, you may want to read up on >> > Cauchy's theorem. Also, the important thing about the Fermi-Dirac >> > distribution, in this case, is just the location of its poles, which >> > should >> > be evident by inspection of the function. >> > >> > Hope this helps, >> > >> > -Jonathan >> > >> > >> > >> > On 06/01/2012 10:09 AM, Juan Manuel Aguiar wrote: >> >> >> >> Dear Users, >> >> Can I assume that nobody understands this issue or knows a reference? >> >> Sincerely >> >> >> >> Juan Manuel >> >> >> >> On Thu, May 31, 2012 at 4:47 PM, Juan Manuel Aguiar >> >> <[email protected]> wrote: >> >>> >> >>> Dear Siesta Users, >> >>> I'm trying to understand the particular shape chosen for the contour >> >>> integration for the density matrix in transiesta. I've already read >> >>> the paper where the method is explained [Phys. Rev. B 65, 165401 >> >>> (2002); pag 5, fig 2] but there the contour is only mentioned, not >> >>> explained. I didn't find this contour in the literature I've found >> >>> about contour integration in NEGF calculations. >> >>> >> >>> I think that the answer must be very easy for the well documented >> >>> user, so I ask you to share with me your knowledge or a reference >> >>> where this particular shape for the contour integration is justified. >> >>> >> >>> Regards >> >>> >> >>> Juan Manuel >> > >> > > >
