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Quantum Boltzmann Transport Equation in the Presence of Interactions; Application to the 2D Hall Effect

C. M. Van Vliet, Th. Samson

Published Oct 1, 1994• 0 citations
Publisher

Abstract

Abstract The quantum Boltzmann equation (QBE), derived in previous papers, is extended to systems with strong interparticle interactions. In particular, the non‐diagonal hierarchy of kinetic equations for [ℒ︁hin]N c cζ, N = 0, 1,…, ∞, where ℒ︁in is the interaction Liouvillian, is solved by iteration in order to obtain a generalized non‐diagonal QBE. The application to the 2D quantum Hall effect, integral and fractional, is discussed.

DOI
10.1002/pssb.2221850215

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Abstract The quantum Boltzmann equation (QBE), derived in previous papers, is extended to systems with strong interparticle interactions. In particular, the non‐diagonal hierarchy of kinetic equations for [ℒ︁hin]N c cζ, N = 0, 1,…, ∞, where ℒ︁in is the interaction Liouvillian, is solved by iteration in order to obtain a generalized non‐diagonal QBE. The application to the 2D quantum Hall effect, integral and fractional, is discussed.
The many-body linear response expressions obtained in previous papers [J. Math. Phys. 19, 1345 (1978); 20, 2573 (1979)] are applied to systems of weakly interacting particles. General expressions for the susceptibility and conductivity in such systems are obtained. The diagonal parts depend on the scattering processes, for which we consider interactions with bosons with mass and electron-phonon interaction. For elastic collisions simple closed forms result. For general two-body collisions, the closed expressions are cumbersome, except when the current is due to collisional current through localized states, such as Landau orbits; in that event a generalized Adams-Holstein result is obtained. The nondiagonal electrical conductivity is shown to be of paramount importance for the quantum mechanical Hall effect. We also derive quantum mechanical Boltzmann equations, both for the diagonal occupancy operator 〈nζ〉t and for the nondiagonal operator 〈c+ζ′ cζ\〉t. The total Boltzmann equation is shown to be fully equivalent with the linear response results. Finally, in the last part we derive the Boltzmann equation for the Wigner function of inhomogeneous systems. In the classical limit this yields the usual Boltzmann transport equation. This equation has therefore been obtained by first principles from the von Neumann equation.
The quantum Boltzmann equation (QBE), originally formulated as the quantum extension of Boltzmann’s classical transport theory by Uehling and Uhlenbeck and subsequently developed within the nonequilibrium Green-function formalisms of Kadanoff–Baym and Keldysh, provides the central kinetic framework for describing relaxation and transport in weakly interacting quantum many-body systems. Although its formal structure is well established, conventional derivations often obscure the microscopic equivalence between operator and nonequilibrium Green-function approaches. Wick factorization is typically applied without explicitly enumerating real-time contractions, while Green-function treatments encode the same physics in Born-level self-energies that conceal their operator origin. In this paper, we present a pedagogical microscopic derivation of the quantum Boltzmann equation that makes this equivalence fully explicit. Starting from the ladder-operator formulation, we enumerate all fifteen Wick pairings of the second-order six-field correlator, identifying three Hartree–Fock contributions and twelve real-time Born scattering channels generated by a quartic, number-conserving interaction Hamiltonian. We then establish a one-to-one correspondence between these twelve operator contractions and the Keldysh components of the second-order Born self-energy, demonstrating explicitly how the Uehling–Uhlenbeck collision integral emerges after Wigner transformation and quasiparticle projection. Under controlled Born–Markov and quasiparticle approximations, this work offers a transparent synthesis of operator and Green-function formulations, and clarifies how kinetic fluctuations can be understood as emergent under coarse-graining.
We derive the quantum Boltzmann equation (QBE) of composite fermions at/near the \ensuremath{\nu}=1/2 state using the nonequilibrium Green's-function technique. The lowest-order perturbative correction to the self-energy due to the strong gauge-field fluctuations suggests that there is no well-defined Landau quasiparticle. Therefore, we cannot assume the existence of the Landau quasiparticles a priori in the derivation of the QBE. Using an alternative formulation, we derive the QBE for the generalized Fermi-surface displacement which corresponds to the local variation of the chemical potential in momentum space. From this QBE, one can understand in a unified fashion the Fermi-liquid behaviors of the density-density and the current-current correlation functions at \ensuremath{\nu}=1/2 (in the long-wavelength and the low-frequency limits) and the singular behavior of the energy gap obtained from the finite-temperature activation behavior of the compressibility near \ensuremath{\nu}=1/2. Implications of these results for recent experiments are also discussed.
A description of quantum transport for electrons in the presence of a random impurity potential is provided in terms of a kinetic equation, which is obtained from the diagrammatic structure for the linear response functions. When the electron-electron interaction is neglected, the kinetic equation is of the Boltzmann type, with an effective scattering kernel that signals localization effects. With the further inclusion of the interaction, a suitable distribution function for the energy of the quasiparticles can be introduced even in the presence of strong disorder. In this case the kinetic equation has the diffusive form, with diffusion coefficient and Landau interaction parameters that get renormalized by the interplay of disorder and interaction. The quasiparticles kinetic equation is also shown to contain all known results for the hydrodynamic correlation functions.
By means of the time-smoothing method developed in the previous papers of this series, the quantum kinetic equation is derived from the principles of statistical mechanics. In this paper, we avoid the BBGKY hierarchy; instead we use the Liouville-von Neumann equation for the time-smoothed density matrix. The Boltzmann distribution function is defined in terms of this time-smoothed density matrix. The present scheme may be applied to strong-coupling systems even if they are not in near equilibrium.
In the framework of real-time Green's functions, the general kinetic equations are investigated in a first-order gradient expansion. Within this approximation, the problem of the reconstruction of the two-time correlation functions from the one-time Wigner function was solved. For the Wigner function, a cluster expansion is found in terms of a quasiparticle distribution function. In equilibrium, this expansion leads to the well-known generalized Beth-Uhlenbeck expression of the second virial coefficient. As a special case, the T-matrix approximation for the self-energy is investigated. The quantum kinetic equation derived thus has, besides the (Markovian) Boltzmann collision integral, additional terms due to the retardation expansion which reflect memory effects. Special interest is paid to the case that bound states exist in the system. It is shown that the bound state contribution, which can be introduced via a bilinear expansion of the two-particle T matrix, follows from the first-order retardation term in the general kinetic equation. The full Wigner function is now a sum of one function describing the unbound particles and another one for the bound state contribution. The latter two functions have to be determined from a coupled set of kinetic equations. In contrast to the quantum Boltzmann equation, energy and density of a nonideal system are conserved. \textcopyright{} 1996 The American Physical Society.