A general study of electron linear transport in systems with open boundary conditions is presented. It is shown that a consistent treatment of bulk and boundary contributions is only achieved in a strictly gauge-invariant formulation of the electomagnetic linear response. Invariance under large gauge transformations requires the use of a generalized version of the continuity equation in which current sources at infinity are explicitly included. These sources are described by delta functions located at the points of infinity and they guarantee global charge conservation. The formulations of steady-state transport in terms of global conductance coefficients and nonlocal conductivity tensors are shown to be strictly equivalent for interacting electron systems in the presence of a magnetic field. As a corollary, the reciprocity theorem for a four-lead resistance is proved in its full generality for systems with arbitrary many-body interactions.
A general study of electron linear transport in systems with open boundary conditions is presented. It is shown that a consistent treatment of bulk and boundary contributions is only achieved in a strictly gauge-invariant formulation of the electomagnetic linear response. Invariance under large gauge transformations requires the use of a generalized version of the continuity equation in which current sources at infinity are explicitly included. These sources are described by delta functions located at the points of infinity and they guarantee global charge conservation. The formulations of steady-state transport in terms of global conductance coefficients and nonlocal conductivity tensors are shown to be strictly equivalent for interacting electron systems in the presence of a magnetic field. As a corollary, the reciprocity theorem for a four-lead resistance is proved in its full generality for systems with arbitrary many-body interactions.
We analyze transport in a uniform electric field of arbitrary strength by using a gauge-invariant formulation of the Airy representation of the Baym-Kadanoff-Keldysh theory of nonequilibrium quantum statistical mechanics. For a model electron-phonon interaction, experimentally relevant quantities can be extracted from a function of one variable; this function obeys a linear homogeneous integral equation. The quantum distribution function and the nonlinear drift velocity versus applied field characteristics are evaluated numerically. Many features of our nonlinear theory bear a formal resemblance to linear-response theory.
There exists a regime of overlap between the equilibrium and non-equilibrium behavior of a system, the non-equilibrium behavior of weakly perturbed states. When a system is perturbed ever so slightly, its response will be linear in the perturbation, say the current of the conduction electrons in a metal will be proportional to the strength of the applied electric field. This regime is called the linear response regime, and though the system is in a non-equilibrium state all its characteristics can be inferred from the properties of its equilibrium state. In the next chapter we shall go beyond the linear regime by showing how to obtain quantum kinetic equations. The kinetic-equation approach to transport is a general method, and allows in principle nonlinear effects to be considered. However, in many practical situations one is interested only in the linear response of a system to an external force. The linear response limit is a tremendous simplification in comparison with general non-equilibrium conditions, and is the subject matter of this chapter. In particular the linear response of the density and current of an electron gas are discussed. The symmetry properties of response functions, and the fluctuation–dissipation theorem are established. Lastly we demonstrate how correlation functions can be measured in scattering experiments, as illustrated by considering neutron scattering from matter. Needless to say, in measurements of (say) the current in a macroscopic body, far less information in the current correlation function is probed. Linear response In this section we consider the response of an arbitrary property of a system to a general perturbation.
Even though its classical equations of motion are then left invariant, when an action is redefined by an additive total derivative or divergence term (in time, in the case of a mechanical system) such a transformation induces non-trivial consequences for the system’s canonical phase space formulation. This is even more true and then in more subtle ways for the canonically quantized dynamics, with, in particular, an induced transformation in the unitary configuration space representation of the Heisenberg algebra being used for the quantum system. When coupled to a background gauge field, such considerations become crucial for a proper understanding of the consequences for the system’s quantum dynamics of gauge transformations of that classical external background gauge field, while under such transformations, the system’s degrees of freedom, abstract quantum states and quantum dynamics are certainly strictly invariant. After a detailed analysis of these different points in a general context, these are then illustrated specifically in the case of the quantum Landau problem with its classical external background magnetic vector potential for which the most general possible parametrized gauge choice is implemented herein. The latter discussion aims as well to clarify some perplexing statements in the literature regarding the status of gauge choices to be made for the magnetic vector potential for that quantum system. The role of the global spacetime symmetries of the Landau problem and their gauge invariant Noether charges is then also emphasized.
Gauge-invariant Wigner theory describes the quantum-mechanical evolution of charged particles in the presence of an electromagnetic field in phase space, which is spanned by position and kinetic momentum. This approach is independent of the chosen potentials, as it depends only on the electric and magnetic field variables. Several approaches to derive a gauge-invariant Wigner evolution equation have been reported, which are generally complex. This work presents a new formulation for a single electron in a general electromagnetic field based solely on differential operators that simplify existing formulations. A gauge-dependent equation is derived first using Moyal's equation. A transformation of the Wigner function, introduced by Stratonovich, is then used to make the equation gauge-invariant, which gives us a strong formulation of the problem. This equation can be transformed into its weak form, which proves that both formulations are equivalent. An analysis of the different properties of the gauge-dependent and gauge-invariant formulations is given, as well as the different requirements for the regularity and asymptotic behavior of the strong and weak formulations.
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 main purpose of this paper is to further our theoretical understanding of the fractional quantum Hall effect, in particular of spin effects, in two-dimensional incompressible electron fluids subject to a strong, transverse magnetic field. As a prerequisite for an analysis of the quantum Hall effect, the authors develop a general formulation of the many-body theory of spinning particles coupled to external electromagnetic fields and moving through a general, geometrically nontrivial background. Their formulation is based on a Lagrangian path-integral quantization and is valid in arbitrary coordinates, including coordinates moving according to a volume-preserving flow. It is found that nonrelativistic quantum theory exhibits a fundamental, local U(1)\ifmmode\times\else\texttimes\fi{}SU(2) gauge invariance, and the corresponding gauge fields are identified with physical, external fields. To illustrate the usefulness of their formalism, the authors prove a general form of the quantum-mechanical Larmor theorem and discuss some well-known effects, including the Barnett-Einstein-de Haas effect and superconductivity, emphasizing the implications of U(1)\ifmmode\times\else\texttimes\fi{}SU(2) gauge invariance. They then consider two-dimensional, incompressible quantum fluids in more detail. Exploiting U(1)\ifmmode\times\else\texttimes\fi{}SU(2) gauge invariance, they calculate the leading terms in the effective actions of such systems as functionals of the U(1) and SU(2) gauge fields, on large-distance and low-frequency scales. Among the applications of these results are a simple proof of the Goldstone theorem for spin waves and the linearresponse theory of two-dimensional, incompressible Hall fluids, including a Hall effect for spin currents and sum rules for the response coefficients. For two-dimensional, incompressible systems with broken parity and time-reversal symmetry, a particularly significant implication of U(1)\ifmmode\times\else\texttimes\fi{}SU(2) gauge invariance is a duality between the physics inside the bulk of the system and the physics of gapless, chiral modes propagating along the boundary of the system. These modes form chiral $\stackrel{^}{\mathrm{u}}(1)$ and $\mathrm{s}\stackrel{^}{\mathrm{u}}(2)$ current algebras. The representation theory of these current algebras, combined with natural physical constraints, permits one to derive the quantization of the response coefficients, such as the Hall conductivity. A classification of incompressible Hall fluids is outlined, and many examples, including one concerning a superfluid 3He-BA interface, are discussed.