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arXiv:2108.05431 · Physical review. E

Real-time evolution of static electron-phonon models in time-dependent electric fields

Manuel Weber, James K. Freericks

Published Feb 3, 2022• 9 citations
arXivHTMLPDF

Abstract

We present an exact Monte Carlo method to simulate the nonequilibrium dynamics of electron-phonon models in the adiabatic limit of zero phonon frequency. The classical nature of the phonons allows us to sample the equilibrium phonon distribution and efficiently evolve the electronic subsystem in a time-dependent electromagnetic field for each phonon configuration. We demonstrate that our approach is particularly useful for charge-density-wave systems experiencing pulsed electric fields, as they appear in pump-probe experiments. For the half-filled Holstein model in one and two dimensions, we calculate the out-of-equilibrium response of the current and the energy after a pulse is applied as well as the photoemission spectrum before and after the pump. Finite-size effects are under control for chains of 162 sites (in one dimension) or 16×16 square lattices (in two dimensions).

DOI
10.1103/physreve.105.025301

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We present an exact Monte Carlo method to simulate the nonequilibrium dynamics of electron-phonon models in the adiabatic limit of zero phonon frequency. The classical nature of the phonons allows us to sample the equilibrium phonon distribution and efficiently evolve the electronic subsystem in a time-dependent electromagnetic field for each phonon configuration. We demonstrate that our approach is particularly useful for charge-density-wave systems experiencing pulsed electric fields, as they appear in pump-probe experiments. For the half-filled Holstein model in one and two dimensions, we calculate the out-of-equilibrium response of the current and the energy after a pulse is applied as well as the photoemission spectrum before and after the pump. Finite-size effects are under control for chains of 162 sites (in one dimension) or 16×16 square lattices (in two dimensions).
Monte Carlo simulations are performed to examine superconductivity and charge-density-wave fluctuations in the infinite-dimensional electron-phonon Holstein model. The electron-phonon system interpolates between an attractive, static, Falicov-Kimball model that always exhibits charge-density-wave order and an attractive, instantaneous, Hubbard model that always superconducts as a function of phonon frequency. The maximum charge-density-wave transition temperature at half-filling is an order of magnitude smaller than the effective electronic bandwidth and is virtually independent of the phonon frequency. The maximum superconducting transition temperature depends strongly on phonon frequency and is bounded from above by the maximum charge-density-wave transition temperature. The exact solution is compared to both weak-coupling expansions and strong-coupling expansions. The effective phonon potential becomes anharmonic and develops a double-well structure that deepens as the electron-phonon interaction increases.
The spectral response and physical features of the 2D Hubbard-Holstein model are calculated both in equilibrium at zero and low chemical dopings, and after an ultrashort powerful light pulse, in undoped systems. At equilibrium and at strong charge-lattice couplings, the optical conductivity reveals a three-peak structure in agreement with experimental observations. After an ultrashort pulse and at nonzero electron-phonon interaction, phonon and spin subsystems oscillate with the phonon period T(ph)≈80 fs. The decay time of the phonon oscillations is about 150-200 fs, similar to the relaxation time of the charge system. We propose a criterion for observing these oscillations in high T(c) compounds: the time span of the pump light pulse τ(pump) has to be shorter than the phonon oscillation period T(ph).
We derive frequency moment sum rules for the retarded electronic Green's function and self-energy of the Holstein model for both equilibrium and nonequilibrium cases. We also derive sum rules for the phonon propagator in equilibrium and nonequilibrium. These sum rules allow one to benchmark nonequilibrium calculations and help with interpreting the behavior of electrons driven out of equilibrium by an applied electric field. We exactly evaluate the sum rules when the system is in the atomic limit. We also discuss the application of these sum rules to pump/probe experiments like time-resolved angle-resolved photoemission spectroscopy.
We derive frequency moment sum rules for the retarded electronic Green's function and self-energy of the Holstein model for both equilibrium and nonequilibrium cases. We also derive sum rules for the phonon propagator in equilibrium and nonequilibrium. These sum rules allow one to benchmark nonequilibrium calculations and help with interpreting the behavior of electrons driven out of equilibrium by an applied electric field. We exactly evaluate the sum rules when the system is in the atomic limit. We also discuss the application of these sum rules to pump/probe experiments like time-resolved angle-resolved photoemission spectroscopy.
In this dissertation a newly developed numerical method is presented, which is optimized for the time evolution of one-dimensional lattice systems with large local Hilbert spaces. This method extends the time-evolving block-decimation (TEBD) to include a local basis optimization (LBO), which has already been successfully combined with ground state methods. The algorithm is based on matrix product states (MPS), which can represent the quantum state of a one-dimensional chain in most cases with a number of parameters, that is not exponentially increasing with the chain length. The LBO causes a reduction of the simulation times that is linear in the bond dimension of the MPS. To demonstrate the advantages of this method, we apply the TEBD-LBO to electron-phonon (e-p) systems. In this thesis these are described by the Holstein model, which goes beyond semi-classical approximations and covers the full quantum statistics of the phonons. The understanding of the nonequilibrium dynamics of charge carriers coupled to lattice vibrations is of great importance for research areas like transport through quasi one-dimensional conductors, photo-generated phase transitions and time-resolved spectroscopy. First, the energy transfer from a highly excited electron to the phononic degrees of freedom on a small chain is studied. In the various parameter regimes different types of relaxation occur. In any case, after a certain time the system reaches a state, where on average no energy is exchanged between the electron and phonons. This can either mean a constant kinetic energy or oscillations with a constant amplitude and frequency. Next, long perfectly conducting leads without coupling to phonons were attached on both sides of the small chain. In the left lead an electron with density distribution in the shape of a Gaussian wave packet is injected with momentum towards the e-p coupled structure in the middle. This structure acts as an impurity in an otherwise perfectly conducting chain. The investigation shows resonance effects in the transmission and reflection at this impurity. Further, the electron can transfer a part of its energy permanently to the phonons, which results in a reduction of the velocity. Finally, two mechanisms are presented that lead to self-trapping of the electron on the e-p coupled structure.
A nonperturbative technique is developed to solve for the quantum dynamics of an electron interacting strongly with its environment. Real-time path-integral methods are used to solve numerically a quantum dissipation problem that simulates nonequilibrium electron transport in solids. A linearly coupled electron-phonon interaction is used to model the dissipation. This method is nonperturbative in both the electron-phonon coupling and the external electric field, making it useful for electron-transport calculations in cases of both high fields and large phonon-scattering rates. Monte Carlo sampling over the electron paths is done using the Feynman-Vernon influence functional as a weight. Importance-sampling methods are developed for the numerical solution of the quantum propagation. Results of the calculations are shown which include the motion of the electron, the wave-packet spread, and temperature effects.