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Quantum transport for a many-body system

V.V. Brkovic

Published Nov 19, 2002• 0 citations
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Abstract

We treat the quantum transport of an interacting system of electrons, impurities and phonons. The dynamic system is in a time-dependent electric field. We analyzed quantum transport process by using the quantum generalized Langevin equation, in which the system is shown to be equivalent to a quantum particle in a heat bath. After eliminating the heat-bath variables, the equation of motion for the quantum particle is written in a form of quantum generalized Langevin equation, with a memory term which reflects the retarded effects of the heat bath on the quantum particle.

DOI
10.1109/icmel.1995.500851

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We treat the quantum transport of an interacting system of electrons, impurities and phonons. The dynamic system is in a time-dependent electric field. We analyzed quantum transport process by using the quantum generalized Langevin equation, in which the system is shown to be equivalent to a quantum particle in a heat bath. After eliminating the heat-bath variables, the equation of motion for the quantum particle is written in a form of quantum generalized Langevin equation, with a memory term which reflects the retarded effects of the heat bath on the quantum particle.
We treat the quantum transport of an interacting system of electrons, impurities, and phonons, in a time-dependent electric field, by using the quantum generalized Langevin equation (GLE), in which the system is shown to be equivalent to a quantum particle in a heat bath. We follow here the philosophy of Ford, Lewis, and O'Connell, who have demonstrated the usefulness of the GLE approach to heat-bath problems. The center of mass of the electrons acts like a quantum particle, while the relative electrons and phonons play the role of heat bath. They are coupled through the electron-impurity and electron-phonon interactions. After eliminating the heat-bath variables, the equation of motion for the quantum particle is written in a form of a quantum generalized Langevin equation, with a memory term which reflects the retarded effects of the heat bath on the quantum particle. The evaluation of the memory term immediately leads to a result for the susceptibility from which we can calculate the conductivity directly, in contrast to Kubo-type calculations which require the evaluation of correlation functions as an intermediate step. As a demonstration of the directness of our approach, we show that the usual random-phase-approximation conductivity results are easily derived. In addition, we derive an expression for the memory function, which incorporates higher-order electron-impurity scattering.
Quantum diffusion equations featuring explicitly time-dependent transport coefficients are derived from generalized non-Markovian Langevin equations. Generalized fluctuation-dissipation relations and analytic expressions for calculating the friction and diffusion coefficients in nuclear processes are obtained. The asymptotic behavior of the transport coefficients and correlation functions for a damped harmonic oscillator that is linearly coupled in momentum to a heat bath is studied. The coupling to a heat bath in momentum is responsible for the appearance of the diffusion coefficient in coordinate. The problem of regression of correlations in quantum dissipative systems is analyzed.
The process of diffusion constitutes the dissipative motion of a classical particle or group of particles subjected to many-body interactions with their surroundings. When the surroundings consist of an infinitely large system that can rapidly recover from any deviation from thermal equilibrium such that its temperature remains roughly uniform, the system qualifies as a heat bath. Further, when the degrees of freedom of the heat bath are integrated out because of the implied timescale separation, the dynamics of the subsystem appear dissipative, characterized by parameters such as friction coefficient, diffusion constant, and others.
The macroscopic description of a quantum particle with passive dissipation and moving in an arbitrary external potential is formulated in terms of the generalized Langevin equation. The coupling with the heat bath corresponds to two terms: a mean force characterized by a memory function \ensuremath{\mu}(t) and an operator-valued random force. Explicit expressions are given for the correlation and commutator of the random force. The random force is never Markovian. It is shown that \ensuremath{\mu}\ifmmode \tilde{}\else \~{}\fi{}(z), the Fourier transform of the memory function, must be a positive real function, analytic in the upper half-plane and with Re[\ensuremath{\mu}\ifmmode \tilde{}\else \~{}\fi{}(\ensuremath{\omega}+i0+{0}^{+}0+)] a positive distribution on the real axis. This form is then derived for the independent-oscillator model of a heat bath. It is shown that the most general quantum Langevin equation can be realized by this simple model. A critical comparison is made with a number of other models that have appeared in the literature.
The second-quantized, many-electron, atomic, and molecular Hamiltonian is partitioned both by the identity or labeling of the spin orbitals and by the dynamics of the spin orbitals into a system coupled to a bath. The electron bath is treated by a molecular time scale generalized Langevin equation approach designed to include one-electron dynamics in the system dynamics. The bath is formulated as an equivalent chain of spin orbitals through the introduction of equivalent-chain annihilation and creation operators. Both the dynamics and the quantum grand canonical statistical properties of the electron bath are examined. Two versions for the statistical properties of the bath are pursued. Using a weak bath assumption, a bath statistical average is defined which allows one to achieve a reduced dynamics description of the electron system which is coupled to the electron bath. In a strong bath assumption effective Hamiltonians are obtained which reproduce the dynamics of the bath and which lead to the same results as found in the weak bath assumption. The effective (but exact) Hamiltonian is found to be a one-electron Hamiltonian. A reduced dynamics equation of motion for the system population matrix is derived and found to agree with a previous version. This equation of motion is useful for studying electron transfer in the system when coupled to an electron bath.
The generalized Langevin equation and its attendant fluctuation–dissipation relation (FDR) for both classical and quantum systems is explictly derived for a large class of system-bath interaction potentials. We demonstrate for this class of potentials that the classical FDR involving only the temperature of the bath is satisfied, and that in general the decay times of the dissipative processes and of the system are temperature dependent. We also demonstrate that the quantum FDR depends in detail on the nature of the bath and on the specific system-bath interaction. Thus we conclude that while the classical Langevin equation is phenomenologically useful, its quantum counterpart is much more limited.