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Physical review. B, Condensed matter

Quantum transport for a many-body system using a quantum Langevin-equation approach

G. Y. Hu, Robert F. O’Connell

Published Oct 15, 1987• 52 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.36.5798

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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.
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.
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.
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.
In a paper with the same title as the above, Manoliu and Kittel calculated the high temperature ensemble average of the nonequal time product of the displacement and random force in the classical Brownian motion problem. Here, using a generalized quantum Langevin equation (GLE), we generalize these considerations to the quantum case and to the case of an arbitrary heat bath (such as, e.g., the blackbody radiation heat bath). Also, we consider an arbitrary temperature and we generalize beyond Brownian motion to the case of a harmonically bound particle in a heat bath. In addition, we point out that correlations of the type considered are widely used in many applications in a variety of different areas of physics and we review some recent applications of the GLE.
The effect of an electron heat bath on the motion of an electron system is studied using a quantum Langevin equation and deriving the influence functional of the Feynman-Vernon approach. The generating functional for the time-dependent, finite-temperature correlation functions of a disordered electron system with interactions is written down, and some of its symmetry properties are analyzed. Since the disorder only affects the time evolution of the system our formalism avoids the replica trick and provides a natural physical framework for studying a large variety of open problems in the field of disordered systems. As an example, we give an explicit expression for the averaged generating functional for a disordered electron system in the presence of a Coulomb interaction in terms of two complex matrix fields.
In a recent paper, Ford, Lewis, and O'Connell [Phys. Rev. A 37, 4419 (1988)] considered a charged quantum particle moving in an arbitrary potential and linearly coupled to a heat bath, and they showed that the macroscopic equation describing the time development of the particle motion is in the form of a quantum generalized Langevin equation. We generalize these results to include the presence of an external magnetic field. We find that the magnetic field manifests itself in the presence of an additional term in the Langevin equation, which is the quantum generalization of the Lorentz force, but the magnetic field does not affect the memory function nor the random force appearing in the quantum Langevin equation. It follows that the noise-noise autocorrelation function, as well as the nonequal time commutator of the noise, is the same as that in the absence of a magnetic field. The case of a blackbody radiation heat bath is shown to be easily analyzed as a special case of our general formalism.