Non-equilibrium Green's functions and Current in tkwant
Hello, The kwant documentation on kwant.solvers.common.GreensFunction() states that calling this method will return the retarded Green's function. Does this method evaluate the equilibrium or non-equilibrium retarded Green's function? I am interested in the time-dependent non-equilibrium density matrix, which can be written as the kinetic lesser NE Green's function G_<(t,t') rho_neq (t) = -iG_<(t,t')|t=t'. Thank you in advance. Cheers, Rudolf
Hi Rudolf,
The kwant documentation on kwant.solvers.common.GreensFunction() states that calling this method will return the retarded Green's function. Does this method evaluate the equilibrium or non-equilibrium retarded Green's function? I am interested in the time-dependent non-equilibrium density matrix, which can be written as the kinetic lesser NE Green's function G_<(t,t')
rho_neq (t) = -iG_<(t,t')|t=t'.
'kwant.greens_function' essentially gives you a few matrix elements of the following quantity: Gᴿ(E) = (H + Σ(E))⁻¹ where Σ is the self-energy of the leads at energy E. Specifically it gives you the matrix elements that connect the interface sites of the different leads; these are the matrix elements needed to calculate transport quantities (like conductance). In order to calculate the lesser Green's function for the time-independent, nonequilibrium case (i.e. different chemical potentials / temperatures in different leads) you would need to take a different approach I think. One way to do so would be to apply eq. 22 of https://arxiv.org/abs/1307.6419 to the special case where Ψ_αE(t) has trivial time dependence: Ψ_αE(t) = exp(-iEt) Ψ_αE where Ψ_αE is the scattering state corresponding to incoming mode α at energy E (i.e. what kwant.wave_function gives you). If your system is *time-dependent* then you will need to calculate the time-evolved scattering states, which Kwant cannot give you directly. Indeed I did some work on calculating such quantities during my PhD., and one of the outputs was a code "tkwant", which is essentially a solver for working with Kwant systems that include time-dependence. It essentially does the time evolution and energy integration for you. tkwant is still actively developed, but I am no longer directly involved; you can check it out here: https://gitlab.kwant-project.org/kwant/tkwant/ there's also links to the documentation which includes some examples which should help you get started if this is the direction you want to go. Happy (t)kwanting, Joe
participants (2)
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Joseph Weston
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Rudolf