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arXiv:2307.12543 · Journal of the Physical Society of Japan

Energy Flow during Relaxation in an Electron–Phonon System with Multiple Modes: A Nonequilibrium Green’s Function Study

Ken Inayoshi, Akihisa Koga, Yuta Murakami

Published Nov 1, 2023• 1 citation
arXivHTMLPDF

Abstract

We investigate an energy flow in an extended Holstein model describing electron systems coupled to hot-phonons and heat-bath phonons. To analyze the relaxation process after the photo-excitation of electrons, we employ the nonequilibrium dynamical mean-field theory (DMFT). We find the backward energy flow during the relaxation, where the direction of energy transfer between electrons and hot-phonons is reversed. To clarify the microscopic mechanism of the backward energy flow, we introduce the approximated energy flows, which are calculated with the gradient and quasiparticle approximations and are related to the nonequilibrium distribution functions. We compare these approximated energy flows with the full energy flows calculated from the nonequilibrium DMFT. We find that, in the weak electron-hot-phonon coupling regime, the full and approximated energy flows are almost the same, meaning that the relaxation dynamics can be correctly understood in terms of the nonequilibrium distribution functions. As the strength of the electron-hot-phonon coupling increases, the approximated energy flow fails to qualitatively reproduce the full energy flow. This indicates that the microscopic origin of the energy flow cannot be solely explained by the nonequilibrium distribution functions. By comparing the energy flows with different levels of approximation, we reveal the role of the gradient and quasiparticle approximations.

DOI
10.7566/jpsj.92.124001

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We investigate an energy flow in an extended Holstein model describing electron systems coupled to hot-phonons and heat-bath phonons. To analyze the relaxation process after the photo-excitation of electrons, we employ the nonequilibrium dynamical mean-field theory (DMFT). We find the backward energy flow during the relaxation, where the direction of energy transfer between electrons and hot-phonons is reversed. To clarify the microscopic mechanism of the backward energy flow, we introduce the approximated energy flows, which are calculated with the gradient and quasiparticle approximations and are related to the nonequilibrium distribution functions. We compare these approximated energy flows with the full energy flows calculated from the nonequilibrium DMFT. We find that, in the weak electron-hot-phonon coupling regime, the full and approximated energy flows are almost the same, meaning that the relaxation dynamics can be correctly understood in terms of the nonequilibrium distribution functions. As the strength of the electron-hot-phonon coupling increases, the approximated energy flow fails to qualitatively reproduce the full energy flow. This indicates that the microscopic origin of the energy flow cannot be solely explained by the nonequilibrium distribution functions. By comparing the energy flows with different levels of approximation, we reveal the role of the gradient and quasiparticle approximations.
We investigate the nonequilibrium dynamics of the weak-coupling Hubbard-Holstein model after a sudden switch-on of the electron-phonon interaction within nonequilibrium dynamical mean-field theory (DMFT). Using the self-consistent Migdal approximation for the electron-phonon coupling together with second-order perturbation theory for the electron-electron interaction, we show that the relaxation dynamics exhibits a crossover between electron-dominated and phonon-dominated regimes, extending to finite Hubbard interaction the scenario previously identified in the Holstein model. To investigate the microscopic buildup of the thermal state, we analyze the dynamics within the Step-by-Step DMFT framework. In the plane of real time and DMFT iteration number, thermalization is marked by a sharp propagating front. This front appears in electronic observables already for weak quenches within the simulated time window, whereas the phononic sector exhibits a visible front only at sufficiently strong coupling. Thus, at weak coupling the local dispersionless phonons show a delayed onset of front formation, while near and beyond the crossover the front develops on comparable timescales in both the electronic and phononic sectors. Whenever both fronts are resolved, they propagate with the same velocity, showing that thermalization spreads coherently through the coupled electron-phonon system.
We study the relaxation of the Holstein model after a sudden switch-on of the interaction by means of the nonequilibrium dynamical mean field theory, with the self-consistent Migdal approximation as an impurity solver. We show that there exists a qualitative change in the thermalization dynamics as the interaction is varied in the weak-coupling regime. On the weaker interaction side of this crossover, the phonon oscillations are damped more rapidly than the electron thermalization time scale, as determined from the relaxation of the electron momentum distribution function. On the stronger interaction side, the relaxation of the electrons becomes faster than the phonon damping. In this regime, despite long-lived phonon oscillations, a thermalized momentum distribution is realized temporarily. The origin of the ``thermalization crossover'' found here is traced back to different behaviors of the electron and phonon self-energies as a function of the electron-phonon coupling. In addition, the importance of the phonon dynamics is demonstrated by comparing the self-consistent Migdal results with those obtained with a simpler Hartree-Fock impurity solver that neglects the phonon self-energy. The latter scheme does not properly describe the evolution and thermalization of isolated electron-phonon systems.
We introduce a simple ansatz for the wave function of a many-body system based on coupled forward and backward propagating semiclassical trajectories. This method is primarily aimed at, but not limited to, treating nonequilibrium dynamics in electron-phonon systems. The time evolution of the system is obtained from the Euler-Lagrange variational principle, and we show that this ansatz yields Ehrenfest mean-field theory in the limit that the forward and backward trajectories are orthogonal, and in the limit that they coalesce. We investigate accuracy and performance of this method by simulating electronic relaxation in the spin-boson model and the Holstein model. Although this method involves only pairs of semiclassical trajectories, it shows a substantial improvement over mean-field theory, capturing quantum coherence of nuclear dynamics as well as electron-nuclear correlations. This improvement is particularly evident in nonadiabatic systems, where the accuracy of this coupled trajectory method extends well beyond the perturbative electron-phonon coupling regime. This approach thus provides an attractive route forward to the ab initio description of relaxation processes, such as thermalization, in condensed phase systems.
The nonequilibrium interplay of electrons and phonons plays an important role in the thermalization of solids, yet the microscopic picture of transient states and relaxation pathways remains incomplete. Previous nonequilibrium Green's function (NEGF) studies with full two-time dependence were restricted to local phonons and local self-energy approximations, leaving momentum-dependent phonon dynamics largely unexplored. Here, we demonstrate the recently developed quantics-tensor-train (QTT) NEGF framework via large-scale lattice simulations of models with arbitrary phonon dispersions. QTTs provide a memory-efficient representation of full two-time Green's functions, enabling momentum-resolved simulations with full electron-phonon feedback on lattices up to 256x256 sites. Comparing optical and acoustic phonon models, we reveal a hierarchy of relaxation bottlenecks that extends the well-known phonon-window bottleneck effect. For optical phonons, we confirm the main phonon-energy window and uncover an additional reduced window separating momentum-space regions with excess and deficit electronic population. We also identify a separate phonon-thermalization bottleneck rooted in momentum-dependent coupling to the particle-hole continuum. High momentum and frequency resolution makes the phonon--charge correspondence directly visible. For acoustic phonons, the phonon-energy window acquires pronounced momentum dependence dictated by energy and momentum conservation. The reduced window becomes asymmetric, and directional scattering between Brillouin-zone regions produces a persistent bottleneck for low-momentum phonon modes. Our results establish diagrammatic QTT-NEGF simulations as a scalable framework for quantitative nonequilibrium electron-phonon dynamics, overcoming previous lattice-size and propagation-time limitations and providing accurate reference data for time-resolved spectroscopies.
We study quenches of the interaction and electron-phonon coupling parameter in the Hubbard-Holstein model, using nonequilibrium dynamical mean field theory. The calculations are based on a generalized Lang-Firsov scheme for time-dependent interactions or externally driven phonons, and an approximate strong-coupling impurity solver. The interaction quench calculations reveal the phonon-assisted decay of excess doublons, while the quenches of the electron-phonon coupling lead to persistent oscillations of the phonons and to a phonon-enhanced doublon production.
We present a one-dimensional microscopic model of interacting electrons coupled to phonons that allows the exact calculation of the time evolution of the momentum distribution functions starting from a nonequilibrium initial state. The model shows relaxation features known from approximate calculations for more realistic three-dimensional models. The exact results are compared with approximate ones resulting from the use of quantum kinetic equations which avoid the assumption of completed successive scattering processes implicit in simple Boltzmann equations. To renormalize the electronic but not the phonon propagation in the non-Markovian equations yields no improvement uniform in time compared to simpler approximations. Due to the presence of the Fermi sea only the additional renormalization of the phonon propagation leads to a drastic improvement in comparison with the exact results.