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arXiv:2006.14965 · Physical Review Letters

Fast Green’s Function Method for Ultrafast Electron-Boson Dynamics

Daniel Karlsson, Robert van Leeuwen, Y. Pavlyukh, Enrico Perfetto, Gianluca Stefanucci

Published Jul 15, 2021• 73 citations
arXivHTMLPDF

Abstract

The interaction of electrons with quantized phonons and photons underlies the ultrafast dynamics of systems ranging from molecules to solids, and it gives rise to a plethora of physical phenomena experimentally accessible using time-resolved techniques. Green's function methods offer an invaluable interpretation tool since scattering mechanisms of growing complexity can be selectively incorporated in the theory. Currently, however, real-time Green's function simulations are either prohibitively expensive due to the cubic scaling with the propagation time or do neglect the feedback of electrons on the bosons, thus violating energy conservation. We put forward a computationally efficient Green's function scheme which overcomes both limitations. The numerical effort scales linearly with the propagation time while the simultaneous dressing of electrons and bosons guarantees the fulfillment of all fundamental conservation laws. We present a real-time study of the phonon-driven relaxation dynamics in an optically excited narrow band-gap insulator, highlighting the nonthermal behavior of the phononic degrees of freedom. Our formulation paves the way to first-principles simulations of electron-boson systems with unprecedented long propagation times.

DOI
10.1103/physrevlett.127.036402

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The interaction of electrons with quantized phonons and photons underlies the ultrafast dynamics of systems ranging from molecules to solids, and it gives rise to a plethora of physical phenomena experimentally accessible using time-resolved techniques. Green's function methods offer an invaluable interpretation tool since scattering mechanisms of growing complexity can be selectively incorporated in the theory. Currently, however, real-time Green's function simulations are either prohibitively expensive due to the cubic scaling with the propagation time or do neglect the feedback of electrons on the bosons, thus violating energy conservation. We put forward a computationally efficient Green's function scheme which overcomes both limitations. The numerical effort scales linearly with the propagation time while the simultaneous dressing of electrons and bosons guarantees the fulfillment of all fundamental conservation laws. We present a real-time study of the phonon-driven relaxation dynamics in an optically excited narrow band-gap insulator, highlighting the nonthermal behavior of the phononic degrees of freedom. Our formulation paves the way to first-principles simulations of electron-boson systems with unprecedented long propagation times.
This work contains a computationally efficient Green's function scheme for treating correlated electron-boson systems out of equilibrium. The numerical effort scales linearly with the simulation time, thus competing with that of the fastest quantum methods available. On a more fundamental level the proposed scheme fulfills energy balance as well as other fundamental conservation laws, and it can be systematically improved using standard many-body perturbation theory. We demonstrate its accuracy and stability through benchmarks against exact results in the Holstein model. However, our formulation equally applies to molecules and solids for first-principles transient calculations with quantum bosons, such as photons and phonons.
Simulations of interacting electrons and bosons out of equilibrium, starting from first principles and aiming at realistic multiscale scenarios, is a grand theoretical challenge. Here, using the formalism of nonequilibrium Green's functions and relying in a crucial way on the recently discovered time-linear formulation of the Kadanoff-Baym equations, we present a versatile toolbox for the simulation of correlated electron-boson dynamics. A large class of methods are available, from the Ehrenfest to the dressed GDGDGD for the treatment of electron-boson interactions in combination with perturbative, i.e., Hartree-Fock and second-Born, or nonperturbative, i.e., GWGWGW and TTT matrices either without or with exchange effects, for the treatment of the Coulomb interaction. In all cases the numerical scaling is linear in time and the equations of motion satisfy all fundamental conservation laws.
Nonequilibrium dynamics of the open chain Holstein-Hubbard model is studied using the linear time-scaling GKBA+ODE\mathrm{GKBA}+\mathrm{ODE}GKBA+ODE scheme developed in Pavlyukh et al. [Phys. Rev. B 105, 125134 (2022)]. We focus on the set of parameters relevant for photovoltaic materials, i.e., a pair of electrons interacting with phonons at the crossover between the adiabatic and antiadiabatic regimes and at moderately large electron-electron interaction. By comparing with exact solutions for two corner cases, we demonstrate the accuracy of the TTT matrix (in the pppppp channel) and the second-order Fan (GDGDGD) approximations for the treatment of electronic ($e\text{\ensuremath{-}}e$) and electron-phonon (eee-ph) correlations, respectively. The feedback of electrons on phonons is consistently included and is shown to be mandatory for the total energy conservation. When two interactions are simultaneously present, our simulations offer a glimpse into the dynamics of doublons and polarons unveiling the formation, propagation and decay of these quasiparticles, energy redistribution between them and self-trapping of electrons.
We present a time-linear scaling method to simulate open and correlated quantum systems out of equilibrium. The method inherits from many-body perturbation theory the possibility to choose selectively the most relevant scattering processes in the dynamics, thereby paving the way to the real-time characterization of correlated ultrafast phenomena in quantum transport. The open system dynamics is described in terms of an "embedding correlator" from which the time-dependent current can be calculated using the Meir-Wingreen formula. We show how to efficiently implement our approach through a simple grafting into recently proposed time-linear Green's function methods for closed systems. Electron-electron and electron-phonon interactions can be treated on equal footing while preserving all fundamental conservation laws.
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Nonequilibrium dynamics of the open chain Holstein-Hubbard model is studied using the linear time-scaling GKBA+ODE scheme developed in the preceeding paper. We focus on the set of parameters relevant for photovoltaic materials, i.\,e., a pair of electrons interacting with phonons at the cross-over between the adiabatic and anti-adiabatic regimes and at moderately large electron-electron interaction. By comparing with exact solutions for two corner cases, we demonstrate the accuracy of the TTT-matrix (in the pppppp channel) and the second-order Fan (GDGDGD) approximations for the treatment of electronic (eee-eee) and electron-phonon (eee-ph) correlations, respectively. The feedback of electron on phonons is consistently included and is shown to be mandatory for the total energy conservation. When two interactions are simultaneously present, our simulations offer a glimpse into the dynamics of doublons and polarons unveiling the formation, propagation and decay of these quasiparticles, energy redistribution between them and self-trapping of electrons.