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Physical review. B, Condensed matter

Hot electron relaxation: Exact solution for a many-electron model

Kurt Schönhammer, C. Wöhler

Published May 15, 1997• 11 citations
Publisher

Abstract

The exact nonequilibrium time evolution of the momentum distribution for a finite many-particle system in one dimension with a linear energy dispersion coupled to optical phonons is presented. For distinguishable particles the influence function of the phonon bath can be evaluated also for a finite particle density in the thermodynamic limit. In the case of fermions the exact fulfillment of the Pauli principle involves a sum over permutations of the electrons and the numerical evaluation is restricted to a finite number of electrons. In the dynamics the antisymmetry of the wave function shows up in the obvious Pauli blocking of momentum states as well as more subtle interference effects. The model shows the expected physical features known from approximate treatments of more realistic models for the relaxation in the energy regime far from the bottom of the conduction band and provides an excellent testing ground for quantum kinetic equations.

DOI
10.1103/physrevb.55.13564

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The exact nonequilibrium time evolution of the momentum distribution for a finite many-particle system in one dimension with a linear energy dispersion coupled to optical phonons is presented. For distinguishable particles the influence function of the phonon bath can be evaluated also for a finite particle density in the thermodynamic limit. In the case of fermions the exact fulfillment of the Pauli principle involves a sum over permutations of the electrons and the numerical evaluation is restricted to a finite number of electrons. In the dynamics the antisymmetry of the wave function shows up in the obvious Pauli blocking of momentum states as well as more subtle interference effects. The model shows the expected physical features known from approximate treatments of more realistic models for the relaxation in the energy regime far from the bottom of the conduction band and provides an excellent testing ground for quantum kinetic equations.
The exact nonequilibrium time evolution of the momentum distribution for a finite many particle system in one dimension with a linear energy dispersion coupled to optical phonons is presented. For distinguishable particles the influence function of the phonon bath can be evaluated also for a finite particle density in the thermodynamic limit. In the case of fermions the exact fulfillment of the Pauli principle involves a sum over permutations of the electrons and the numerical evaluation is restricted to a finite number of electrons. In the dynamics the antisymmetry of the wavefunction shows up in the obvious Pauli blocking of momentum states as well as more subtle interference effects. The model shows the expected physical features known from approximate treatments of more realistic models for the relaxation in the energy regime far from the bottom of the conduction band and provides an excellent testing ground for quantum kinetic equations.
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.
Quantum kinetic equations (QKE) are used for the description of highly excited electrons in the conduction band of semiconductors in the short time regime. The relaxation process is described with QKE in Born approximation and with QKE including two particle correlations. The results of the approximations are compared for different particle densities with the exact solution of a one-dimensional model of N electrons with linear dispersion interacting with optical phonons. Especially the discussion of the processes contributing to the gain peak allows a better understanding of the approximations.
We study the dynamics of relaxation and thermalization in an exactly solvable model of a particle interacting with a harmonic oscillator bath. Our goal is to understand the effects of non-Markovian processes on the relaxational dynamics and to compare the exact evolution of the distribution function with approximate Markovian and non-Markovian quantum kinetics. There are two different cases that are studied in detail: (i) a quasiparticle (resonance) when the renormalized frequency of the particle is above the frequency threshold of the bath and (ii) a stable renormalized "particle" state below this threshold. The time evolution of the occupation number for the particle is evaluated exactly using different approaches that yield to complementary insights. The exact solution allows us to investigate the concept of the formation time of a quasiparticle and to study the difference between the relaxation of the distribution of bare particles and that of quasiparticles. For the case of quasiparticles, the exact occupation number asymptotically tends to a statistical equilibrium distribution that differs from a simple Bose-Einstein form as a result of off-shell processes whereas in the stable particle case, the distribution of particles does not thermalize with the bath. We derive a non-Markovian quantum kinetic equation which resums the perturbative series and includes off-shell effects. A Markovian approximation that includes off-shell contributions and the usual Boltzmann equation (energy conserving) are obtained from the quantum kinetic equation in the limit of wide separation of time scales upon different coarse-graining assumptions. The relaxational dynamics predicted by the non-Markovian, Markovian, and Boltzmann approximations are compared to the exact result. The Boltzmann approach is seen to fail in the case of wide resonances and when threshold and renormalization effects are important.
The study of the interplay between interaction-induced correlations and nonequilibrium initial conditions in many-body systems has recently attracted a lot of attention. New experimental techniques provide high control over many-body systems and systematic access to their nonequilibrium regime. In many systems the nonequilibrium properties of a Fermi liquid can be relevant. A first approach to their understanding is the main content of this thesis. At the beginning, nonequilibrium phenomena, Landau's theory of a Fermi liquid, the Hubbard model, time evolution in quantum mechanics, related experiments, and the flow equation method are reviewed. The key observation of this thesis, namely a characteristic mismatch of expectation values in equilibrium and nonequilibrium, is first illustrated for the squeezed oscillator. For this one-particle model Hamiltonian the perturbative approach can be compared with the exact solution and it is shown that the mismatch holds even beyond perturbation theory. Afterwards these observations are generalized to a larger class of one-particle models. Then the nonequilibrium behavior of a Fermi liquid is examined by analyzing the Fermi liquid phase of the Hubbard model in more than one dimension. After a sudden switch-on of a weak two-particle interaction to the noninteracting Fermi gas the relaxation of the many-body system is observed. For this purpose, the flow equation transformation is implemented for the Hubbard Hamiltonian. Then the discussion of the momentum distribution function and of the kinetic energy displays a three-step relaxation behavior of the Fermi liquid from the initial perturbation until thermalization is reached. Firstly, the sudden switch inserts excitation energy into the system which drives the following dynamics. Then a rapid initial build-up of correlations is caused by initial dephasing and leads to the establishment of a quasiparticle picture. By that the system enters into a quasi-stationary state which can be long-lasting for weak interaction. This state shows prethermalization of the kinetic energy which already has relaxed to its final values. However, the momentum distribution still resembles a zero temperature Fermi liquid. Its later relaxation on a second time scale is caused by a residual two-particle interaction which allows for scattering processes and can be described by a quantum Boltzmann equation. The physical origin of this delayed relaxation can be traced back to an interplay of translational invariance of the Hubbard model and the Pauli principle for fermions; together they restrict the phase space for scattering events. Comparing with similar work I conjecture on the generic nature of the findings made for the quenched Fermi liquid in other many-body systems. Finally I point out to the potential relevance of the delayed relaxation for the observation of further nonequilibrium phenomena, for instance in BCS systems. In order to extend the study of sudden switching to arbitrary switching processes the calculation is repeated using the Keldysh perturbation theory. First evaluations for a linear increase of the interaction strength are given. Extensions of this work are suggested and may motivate further research.
We study thermalization dynamics in a fermion-phonon variant of the Sachdev-Ye-Kitaev model coupled to an external cold thermal bath of harmonic oscillators. We find that quantum critical fermions thermalize more efficiently than phonons, in sharp contrast to the behavior in the Fermi liquid regime. In addition, after a short prethermal stage, the system acquires a quasithermal distribution given by a time-dependent effective temperature, reminiscent of ``hydrodynamic'' relaxation. All physical observables relax at the same rate which scales with the final temperature through an exponent that depends universally on the low-energy spectrum of the system and the bath. Such relaxation rate is derived using a hydrodynamic approximation in full agreement with the numerical solution of a set quantum kinetic equations derived from the Keldysh formalism for nonequilibrium Green's functions. Our results hint toward further research on the applicability of the hydrodynamic picture in the description of the late time dynamics of open quantum systems despite the absence of conserved quantities in regimes dominated by conserving collisions.