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Oxford University Press eBooks

Current density operators

V. Zlatić, R. Monnier

Published May 22, 2014• 0 citations
Publisher

Abstract

This chapter shows how, for noninteracting particles, the charge and energy current density operators are easily obtained from the continuity equation, with the time derivative of the corresponding density operators given by their commutator with the Hamiltonian in the presence of the external fields. This can be generalized to include spin-orbit scattering from heavy impurities, which gives rise to a spin current. For interacting particles, the commutator is very hard to calculate, and the uniform current operators are derived from the rate of change of the charge and energy polarization operators, which are defined as the first moments of the charge and energy densities, respectively. This is done for continuous models, as well as for the Hubbard, periodic Anderson, and Falicov-Kimball lattice models.

DOI
10.1093/acprof:oso/9780198705413.003.0011

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Royal Society of Chemistry eBooks•Jun 13, 2022•1 citation
This chapter shows how, for noninteracting particles, the charge and energy current density operators are easily obtained from the continuity equation, with the time derivative of the corresponding density operators given by their commutator with the Hamiltonian in the presence of the external fields. This can be generalized to include spin-orbit scattering from heavy impurities, which gives rise to a spin current. For interacting particles, the commutator is very hard to calculate, and the uniform current operators are derived from the rate of change of the charge and energy polarization operators, which are defined as the first moments of the charge and energy densities, respectively. This is done for continuous models, as well as for the Hubbard, periodic Anderson, and Falicov-Kimball lattice models.
The time evolution of the density corresponding to the polarization operator, originally constructed to commute with the Dirac Hamiltonian in the absence of an external electromagnetic field, is investigated in terms of the time-dependent Dirac equation taking the presence of an external electromagnetic field into account. It is found that this time evolution leads to ‘tensorial’ and ‘vectorial’ particle current densities and to the interaction of the spin density with the external electromagnetic field. As the time evolution of the spin density does not refer to a constant of motion (continuity condition) it only serves as auxiliary density. By taking the non-relativistic limit, it is shown that the polarization, spin and magnetization densities are independent of electric field effects and, in addition, no preferred directions can be defined.
The study of electrical currents in a regime where a description as a continuous flow fails and the particle nature becomes important, but where notions such as particle density or currents continue to be meaningful, will be at the heart of the present dissertation. Given such a mesoscopic system through which electrons flow, either actively through quantum pumping or passively due to a potential difference, we investigate quantities such as transferred charge, noise, current fluctuations, and relations between them. More generally we shall examine the statistics of particle transport. All discussions will concentrate on compact devices transferring charge between two one-dimensional reservoirs (the leads) of independent electrons. Firstly, we consider the generating function of the moments of transferred charge and we prove that it is given by the determinant of a single-particle operator. It is formally equivalent to a formula of Lesovik and Levitov, but is shown to be well-defined in the situations under consideration, whereas the former is not, in general. The mechanism leading to regularization is typical for mesoscopic systems: we consider a limiting case that reveals the fundamental structure of the problem. In fact, as suited for an open system, the leads contain an infinite number of charges. It follows mathematically that the second quantization ought to be carried out upon the Fermi sea and not upon the vacuum, a procedure known as the GNS construction. We shall rely heavily on operator algebraic techniques. The set of observables, the C*-algebra of canonical anticommutation relations, admits different representations for systems in the thermodynamic limit, which are described by their reduced one-particle density matrix. An important question will be the existence of operators such as the charge or the time evolution in a given representation, which induce so-called Bogoliubov transformations. These prerequisites at hand, we shall pose the problem of counting statistics clearly, prove that the regularized determinant emerges naturally, and show that it describes the transport of integer charges with non-negative probabilities. Secondly, we introduce a model of instantaneous scattering in which various examples can be computed explicitely. In particular, we show that the above theorem applies for scattering through a smoothly operating pump over a finite time interval. We also exhibit situations where the propagator does exist but the generating function is ambiguous. In a more computational approach, we then turn to the relation of individual moments of transferred charge to current correlators. It turns out that the former can indeed be expressed as integrals of the latter over time, provided the current operators are ordered according to Matthews’ T∗ ordering. It is given by first arranging the charge operators according to their time indices, and only then taking the time derivatives, which induce contact terms whose singular support lies along time coincidences. We shall give two derivations of this result, because the first one is transparent, and the second one is related to contradictory computations in the literature where the system is considered to be measured by an external spin. In the model of constant instantaneous scattering, we then argue from the Lesovik-Levitov determinant that the transport can be interpreted in terms of a binomial process in the long time limit. In a parallel computation, we concentrate on the third cumulant as defined by integration of current correlators and show how the contact terms of the T∗ ordering play an essential role in providing the expected binomial result, too. To close this thesis, we suggest an algebraic framework that lays the foundations for a rigorous discussion of the Casimir effect. The problem of infinites arising in the comparison of two physical situations for a bosonic field is interpreted as the non-equivalence of states on the algebra of canonical commutation relations. We exhibit sufficient conditions on the one-particle Hamiltonians which ensure that the Casimir energy is finite and give an expression for it. We discuss its formal equivalence with the result obtained from the standard mode summing method.
This chapter proves the Jonson-Mahan theorem, which is a generalization of the result for noninteracting systems that all transport coefficients can be derived from integrals over frequency ω of a single transport function multiplied by different powers of ω. The Jonson-Mahan theorem extends this to many-body systems with a local (on-site) interaction, as contained, for example, in the Hubbard, Falicov-Kimball, and periodic Anderson models. This theorem is of great importance, because correlation functions involving the energy current density operator are extremely difficult to obtain by other means for interacting electrons. The proof starts from the observation that they are given by appropriate limits and derivatives of a time-ordered correlation function involving four variables, which are then calculated explicitly.
In this lecture, I lay down the basics for the study of the dynamics of a large collection of identical spin systems by introducing the concept of a density operator and an equation for its evolution. Ensembles of spin systems represent macroscopic samples; in quantum mechanics, these can only be described in statistical terms through a density operator. This latter can be expressed in terms of spin angular momentum operators. The density operator allows all measurable properties of an ensemble of spin systems to be determined.