Hey Joe, > When you say "computing a self-energy corresponds to partially fixing the > order of Gaussian elimination" do you mean that computing the self-energy > forces us to invert Φ, and *this* is what leads to an instability? > As I write this I realise that I am just restating what you said, but I would > like a bit of clarity on this point.
Indeed: when there's a lead bound state at the energy of interest, Φ becomes not invertible. More generally, there are no guarantees on the condition number of Φ. Cheers, Anton > Thanks, > > Joe > > > > -----Original Message----- > From: Anton Akhmerov <anton.akhmerov...@gmail.com> > Sent: Sunday, March 22, 2020 8:41 AM > To: Joseph Weston (Aquent LLC - Canada) <v-jos...@microsoft.com> > Cc: kwant-discuss@kwant-project.org > Subject: [EXTERNAL] Re: [Kwant] Stability of the retarded Green's function > calculation > > Hi Joe, > > Computing the Green's function or the self-energy of the lead fails when a > terminated lead has a bound state at the energy we're looking at (so there's > an exact pole). For example a lead made out of a topological superconductor > always has this problem at zero energy. At the same time the mode solvers > face no singularities in this case. In linear algebra terms computing a > self-energy corresponds to partially fixing the order of Gaussian > elimination, while the most general case may require pivoting incompatible > with this order. > > That's what concerns stability. The speed difference is, I think, > insignificant: the number of right hand sides is smaller with the modes > solver since we use one per incoming mode, not one per degree of freedom. > However in practice obtaining the LU decomposition causes the largest > slowdown. > > Cheers, > Anton > > > On Wed, 18 Mar 2020 at 22:12, Joseph Weston (Aquent LLC - Canada) > <v-jos...@microsoft.com> wrote: > > > > Hello again, > > > > > > > > I Noticed a couple of minor mistakes in my previous email: > > > > > > > > I used the term “out of the box” linear solvers, when I meant “black box” > > linear solvers. By this I meant that we are not tailoring the algorithm > > used for solving the linear system based on the properties of the LHS (e.g. > > as we do in RGF by the choice of scattering region slices), but rather > > trusting that the applied mathematicians have done their job properly, and > > any incidental structure in the LHS is found “automatically” by the solver. > > I claimed that the RHSs for the scattering problem were “indicator vectors > > with 1s in [the] extended part”. This is not true, however the > > interpretation of the RHS as a “single incoming mode” is correct AFAIK. > > > > > > > > Happy Kwanting, > > > > > > > > Joe > > > > > > > > From: Kwant-discuss <kwant-discuss-boun...@kwant-project.org> On > > Behalf Of Joseph Weston (Aquent LLC - Canada) > > Sent: Wednesday, March 18, 2020 12:08 PM > > To: kwant-discuss@kwant-project.org > > Subject: [EXTERNAL] [Kwant] Stability of the retarded Green's function > > calculation > > > > > > > > Hello Kwantoptians, > > > > > > > > I have a question regarding the speed/stability of computing the retarded > > Green’s function of a transport setup using Kwant. > > > > > > > > In the Kwant source-code [1] it is noted that using `kwant.greens_function` > > is “often slower and less stable than the scattering matrix calculation”. I > > was wondering if you could provide me with some references for this > > assertion. > > > > > > > > I have perused the Kwant paper [2] and (part I of) the thesis of Michael > > Wimmer [3] and cannot find any mention of a speed/stability comparison of > > the algorithm implemented by `kwant.greens_function` vs `kwant.smatrix`. > > > > > > > > My understanding is that in both cases a linear system (LHS) is constructed > > and solved for different right hand-sides (RHS) using out of the box linear > > solvers. The solution for each RHS corresponds to one column of the > > retarded Green’s function / extended scattering matrix respectively. The > > difference between `kwant.greens_function` and `kwant.smatrix` is then the > > following. In the former case the leads are taken into account by added the > > retarded self-energy to the LHS and the RHSs are indicator vectors for the > > sites of the lead/scattering region interface, which corresponds to a > > “unit impulse” boundary condition. In the latter case the linear system is > > “extended” so as to include extra unknowns that correspond to the > > scattering amplitudes, and the RHSs are indicator vectors with the 1’s in > > this “extended” part, which corresponds to a “single incoming mode” > > boundary condition. > > > > > > > > It seems to me that the salient difference is in the boundary conditions, > > and I do not have a good intuition as to why one set of boundary conditions > > would make the linear system easier/harder to solve. > > > > > > > > Happy Kwanting! > > > > > > > > Joe > > > > > > > > [1]: > > https://nam06.safelinks.protection.outlook.com/?url=https%3A%2F%2Fgitl > > ab.kwant-project.org%2Fkwant%2Fkwant%2F-%2Fblob%2Fmaster%2Fkwant%2Fsol > > vers%2Fcommon.py%23L428&data=02%7C01%7Cv-josewe%40microsoft.com%7C > > 636fd73dc8614ac7a0c708d7ce776cf0%7C72f988bf86f141af91ab2d7cd011db47%7C > > 1%7C0%7C637204884633045315&sdata=6BCOPJsvwdXfsD2Zi1huK8BjB0avZFaV9 > > rFijI1yUAQ%3D&reserved=0 > > > > [2]: > > https://nam06.safelinks.protection.outlook.com/?url=https%3A%2F%2Fiops > > cience.iop.org%2Farticle%2F10.1088%2F1367-2630%2F16%2F6%2F063065%2Fpdf > > &data=02%7C01%7Cv-josewe%40microsoft.com%7C636fd73dc8614ac7a0c708d > > 7ce776cf0%7C72f988bf86f141af91ab2d7cd011db47%7C1%7C0%7C637204884633045 > > 315&sdata=rYQD5petvR%2FMr18hXSCs9jAhCDBhfewYxAw5XdF3qU8%3D&res > > erved=0 > > > > [3]: > > https://nam06.safelinks.protection.outlook.com/?url=https%3A%2F%2Fepub > > .uni-regensburg.de%2F12142%2F&data=02%7C01%7Cv-josewe%40microsoft. > > com%7C636fd73dc8614ac7a0c708d7ce776cf0%7C72f988bf86f141af91ab2d7cd011db47%7C1%7C0%7C637204884633045315&sdata=lCRUeoPlqwsCreOR01xTCstggBLTPxOPzGOEa22GnOI%3D&reserved=0 > > > > > > > >
