Hello Chi and Ryan, The vecs and vecslmbainv arrays are indeed linear superpositions of the lead modes, which means that the evanescent parts are all mixed. vecs and vecslmbdainv are therefore related by multiplication of an upper-triangular part of the Schur decomposition. Or rather it's a bit more complicated—the propagating modes are diagonalized, the evanescent modes are kept orthogonal (and are therefore superpositions of different translation eigenvectors).
Therefore to extract the wave functions in further unit cells you'll need to dig into the algorithm. In these lines you see the Schur form being reordered to isolate the evanescent modes. You'll want to also keep the Schur form of the eigenproblem. The wave function is given by the Schur vectors times the Schur matrix to the power of the number of the unit cell. Regarding the part with normalization by hopping—that's another tricky bit if your hopping isn't full rank. In general, we project the wave function on the nonzero singular values of the hopping, and define the extract function to obtain the full wave function (see https://gitlab.kwant-project.org/kwant/kwant/-/blob/v1.4.2/kwant/physics/leads.py#L882). Does this give you enough pointers in the right direction? Best, Anton On Fri, 4 Feb 2022 at 11:42, Zhang, Chi <chi.zhang...@imperial.ac.uk> wrote: > > Dear Devs, > > I hope this email finds you well. > > We are two Master's students trying to calculate Andreev/Majorana transition > elements using the kwant boundstate algorithm (doi: > 10.21468/SciPostPhys.4.5.026). For our purposes we need to extract the > evanescent tails of the boundstates in the leads as well, which we had hoped > to obtain from the StabilisedModes object in kwant.InfiniteSystem.modes(). > However, in general, the vecs and vecslmbdainv object are not related by a > constant eigenvalue, as advertised in the documentation. We wonder if you > could tell us how to transform from the vecs and vecslmbdainv objects back to > the eigenvectors*eigenvalues and eigenvectors object for our calculations? > > Many thanks in advance for your response! > > Sincerely, > Chi Zhang & Ryan Tiew > MSci Physics with Theoretical Physics > Imperial College London > > ----------------------------------------------------- > Some technical details of what we have tried here. Please ignore this if you > don't have enough time to read. > > We are aware that kwant is transforming vecslmbdainv by > sqrt(norm(hop))*eigenvectors, and vecs by > (hop/sqrt(norm(hop)))@eigenvectors@eigenvalues. However, this does not seem > to be the full story and we are not able to reproduce what is stored in vecs > and vecslmbdainv when there are more than 1 evanescent modes. In particular, > we have tried to do a real generalised Schur decomposition of the two > matrices in the lead schroedinger's equation, grabbed the vectors stored in > the right unitary, and put them through the basis transformation described > above. Still the there is no sign of better agreements. We therefore we > believe it would be more productive to inquire directly from you how to > transform from vecs and vecslmbdainv to actual eigenvectors and eigenvalues. > Many thanks in advance for your reply.
