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arXiv:1107.3240

Electron random walk in ideal phonon gas. Exact dressed electron density matrix evolution equations

Yu. E. Kuzovlev

Published Jul 16, 2011• 0 citations
arXivHTMLPDF

Abstract

An original approach is suggested to analysis of full quantum Liouville equation for single electron (quantum particle) interacting with ideal phonon gas (harmonic boson thermostat). It is shown that under the thermodynamic limit this equation can be exactly reduced to a system of evolution equations connecting density matrix of the electron and its simplest irreducible correlations with amplitudes of one, two, three and more phonon modes. Possible application of these new equations to explanation of the electron mobility 1/f-type fluctuations in semiconductors and other media is discussed. The special case of electron in static disorder is also considered.

DOI
10.48550/arxiv.1107.3240

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An original approach is suggested to analysis of full quantum Liouville equation for single electron (quantum particle) interacting with ideal phonon gas (harmonic boson thermostat). It is shown that under the thermodynamic limit this equation can be exactly reduced to a system of evolution equations connecting density matrix of the electron and its simplest irreducible correlations with amplitudes of one, two, three and more phonon modes. Possible application of these new equations to explanation of the electron mobility 1/f-type fluctuations in semiconductors and other media is discussed. The special case of electron in static disorder is also considered.
The present paper derives conditions on which the Boltzmann equations for electron-phonon systems (two coupled equations) can be derived without the use of the objectionable random phase approximation. The treatment uses von Neumann's density matrix based on the Hamiltonian introduced some time ago by one of us in connection with superconductivity. Formally exact expressions for the rate of change of coarse-grained one-particle distribution functions are derived. They become identical with the collision terms in the Boltzmann equations if (i) the influence of an external field on them is negligible, (ii a ) particle correlation is absent at an initial time, and (ii b ) the relevant series in terms of the interaction parameter converges and is at all times dominated by a certain term. (ii b ) excludes the possibility of extensive correlations as they occur in superconductivity.
We present a partition-free approach to the evolution of density matrices for open quantum systems coupled to a harmonic environment. The influence functional formalism combined with a two-time Hubbard-Stratonovich transformation allows us to derive a set of exact differential equations for the reduced density matrix of an open system, termed the extended stochastic Liouville--von Neumann equation. Our approach generalizes previous work based on Caldeira-Leggett models and a partitioned initial density matrix. This provides a simple, yet exact, closed-form description for the evolution of open systems from equilibriated initial conditions. The applicability of this model and the potential for numerical implementations are also discussed.
Starting from the quantum statistical master equation derived in [1] we show how the connection to the semi-classical Boltzmann equation (SCBE) can be established and how irreversibility is related to the problem of separability of quantum mechanics. Our principle goal is to find a sound theoretical basis for the description of the evolution of an electron gas in the intermediate regime between pure classical behavior and pure quantum behavior. We investigate the evolution of one-particle properties in a weakly interacting N-electron system confined to a finite spatial region in a near-equilibrium situation that is weakly coupled to a statistical environment. The equations for the reduced n-particle density matrices, with n < N are hierarchically coupled through two-particle interactions. In order to elucidate the role of this type of coupling and of the inter-particle correlations generated by the interaction, we examine first the particular situation where energy transfer between the N-electron system and the statistical environment is negligible, but where the system has a finite memory. We then formulate the general master equation that describes the evolution of the coarse grained one-particle density matrix of an interacting confined electron gas including energy transfer with one or more bath subsystems, which is called the quantum Boltzmann equation (QBE). The connection with phase space is established by expressing the one-particle states in terms of the overcomplete basis of coherent states, which are localized in phase space. In this way we obtain the QBE in phase space. After performing an additional coarse-graining procedure in phase space, and assuming that the interaction of the electron gas and the bath subsystems is local in real space, we obtain the semi-classical Boltzmann equation. The validity range of the classical description, which introduces local dynamics in phase space is discussed.
We consider an open quantum many-particle system in which there are dissipative processes. The evolution of this system is described by a kinetic equation for the density matrix. From the equation describing a random Markov process in this system, we obtain an equation for the single-particle statistical operator. This equation describes the evolution of a system of identical particles in a mean-field approximation. The equation for interacting particles in thermodynamic equilibrium was obtained. The distribution function of a system of interacting electrons in metals has multivalence in a certain region of wave vectors. Among many solutions one is isotropic. Other solutions have the anisotropy of the electron distribution over the wave vectors. The anisotropy arises as a result of repulsion and attraction between electrons.
A general form of a many-body Hamiltonian is considered, which includes an interacting fermionic subsystem coupled to noninteracting extended fermionic and bosonic systems. We show that the exact dynamics of the extended bosonic system can be derived from the reduced density matrix of the subsystem alone, despite the fact that the latter contains information about the subsystem only. The advantage of the formalism is immediately clear: While the reduced density matrix of the subsystem is readily available, the formalism offers access to observables contained in the full density matrix, which is often difficult to obtain. As an example, we consider an extended Holstein model and study the nonequilibrium dynamics of the so-called ``reaction mode'' for different model parameters. The effects of the phonon frequency, the strength of the electron-phonon couplings, and the source-drain bias voltage on the phonon dynamics across the bistability are discussed.
We present a new approach to treat correlations in nonequilibrium quantum many-particle system. The method is based on ideas of configuration interaction theory of exact nonperturbative ground state electronic structure calculations. We use superoperator techniques in Liouville–Fock space and represent the nonequilibrium density matrix as a linear combination of all possible nonequilibrium quasiparticle excitations built on the appropriate reference state. As an example we consider the electron transport through the system with electron–phonon interaction. The concept of embedding (buffer zones between the reservoirs and the correlated quantum system) is used to derive an exact master equation for the reduced density matrix. Using approximate (truncated) expansion of the trial density matrix we obtain the linear system of equations for two-quasiparticle amplitudes. Then we compute the steady-state current and compare the result with other approaches. The current conserving property of the method is proven.