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 > > > > >
