Dear Xiao-Xiao, > Le 26 juil. 2023 à 16:17, Xiaoxiao Zhang <xiaoxiao.zh...@riken.jp> a écrit : > > Dear Xavier, > Thank you very much for your valuable reply. > I actually also found these two papers relevant after a long search in the > literature, mainly because I suspected that tKwant might be much slower in > this case. Your confirmation is very encouraging and now I know these two are > the state of the art if we want to take advantage of the periodicity. > > I understand technically that [2012] is NEGF + general response formulas and > [2018] is static scattering matrix + an energy integral. The crucial energy > integral formula Eq. (7) in [2018] deals with dc current but seems readily > generalizable to any harmonics by doing the Fourier transform of Eq. (6). > So it looks like both methods can deal with any harmonic response. Then the > difference between them puzzles me. > Are they mathematically equivalent, apart from possible technical > differences? (Technically, I suppose [2012] needs to use the less stable and > slower GF solvers in Kwant, which is different from [2018].) > I also noticed that [2018] assumes a sharp ac voltage drop at the lead > interface, which is not mentioned or clear in [2012]. > Yes, both approches look different but are in fact mathematically equivalent. You can find a proof of the equivalence here: https://arxiv.org/abs/1307.6419 <https://arxiv.org/abs/1307.6419>
In practice, I would go with the scattering approach, especially if you’re using Kwant. Best regards, Xavier > Any comment will be appreciated. > > -- > Sincerely > Xiao-Xiao > > From: Xavier Waintal <xavier.wain...@cea.fr> > Date: Tuesday, July 25, 2023 23:33 > To: Xiaoxiao Zhang <xiaoxiao.zh...@riken.jp> > Cc: kwant-discuss@python.org <kwant-discuss@python.org> > Subject: Re: [Kwant] Is Tkwant capable of simulating nonlinear optical > responses > > Hi, > > I see two ways to address this problem > > - You can use Tkwant and indeed do a FFT afterwards. It works but it is > somewhat > overkill and cumbersome. However, one advantage of this approach is that if > you use self-consistent Tkwant, you will be able to take interactions effect > at a level equivalent to RPA which is important for these types > of calculaitons > > - You can use Kwant and one of the many formula that relates its outputs to > finite frequency conductivities. > See e.g. the following references to find them: > > * https://arxiv.org/abs/1802.05924 <https://arxiv.org/abs/1802.05924> > * https://arxiv.org/abs/1211.2768 <https://arxiv.org/abs/1211.2768> > > (this will involve computing an integral over energy of some object > calculable with Kwant). > > Hope it helps, > > Xavier > > > > Le 17 juil. 2023 à 04:20, X.-X. Zhang <xiaoxiao.zh...@riken.jp> a écrit : > > > > Hello Kwant community, > > The Kwant package deals with the (dc) linear response for tight-binding > > models while TKwant is time-dependent. But it is not obvious to me from the > > documentation whether TKwant is capable of simulating nonlinear responses > > (at finite frequencies). Instead of naively assuming that it does not go > > beyond linear Kwant, it is probably worth asking here. > > > > I'm particularly interested in finding nonlinear conductivities like a > > second-harmonic sigma(2w; w, w) and a dc response sigma(0; w, -w) with w > > the driving frequency. They are generated, e.g., by applying the ac bias > > driving voltage/current through the attached leads. If Tkwant does capture > > all such nonlinear effects, I presume one can simulate in the real time and > > transform the current/voltage response to the frequency space and obtain > > these results. > > > > Any relevant comment will be appreciated! >
