Paraxial propagation of a quantum charge in a random magnetic field

Abstract

The paraxial (parabolic) theory of a near forward scattering of a quantum charged particle by a static magnetic field is presented. From the paraxial solution to the Aharonov-Bohm scattering problem the transverse transfered momentum (the Lorentz force) is found. Multiple scattering is considered for two models: (i) Gaussian δ-correlated random magnetic field; (ii) a random array of the Aharonov-Bohm magnetic flux line. The paraxial gauge-invariant two-particle Green function averaged with respect to the random magnetic field is found by an exact evaluation of the Feynman integral. It is shown that in spite of the anomalous character of the forward scattering, the transport properties can be described by the Boltzmann equation. The Landau quantization in the gauge field of the Aharonov-Bohm lines is discussed.

Influence of long-range disorder on electron motion in two dimensions

“…If we keep to first order in all of these terms, we find that the contribution of A coll is (−2) times that of the last term in Eq. (20): in this way we recover the usual form [51,52] of the σ-model for ballistic disorder:…”

“…Consequently a wide variety of alternative techniques have been applied to this problem. A real-space path integral representation has been introduced by Altshuler and Ioffe [16] and Altshuler et al [18], while the eikonal [19] and related paraxial [20] approximations have also been employed. If the correlation length, d, of the random magnetic field is sufficiently large that d ≫ l, 1/k F , where l is the single-particle mean free path and k F is the Fermi momentum, then a "classical" regime is reached in which the contribution of classical memory effects has been shown to be significant [21,22].…”

Magnetic forces in the absence of a classical magnetic field

“…Finally, we would like to mention that Refs. [32] and [33] came to our attention after this manuscript was finished. In those papers, Shelankov uses paraxial analysis to examine some of the same problems we have discussed here, restricted to wave functions of finite width.…”

Tails of the density of states in a random magnetic field

“…In addition, the calculation of the Green function is plagued by infrared divergencies [7,8,9,10,11] that are due to the long-range nature of the correlations of the vector potential, even if the spatial correlations in the RMF are short-ranged. It has been suggested that these divergencies are due to the non-gauge-invariance of the Green function and therefore unphysical [9], although, recently, a physical interpretation has been proposed [11]. In order to avoid such difficulties, E. Altshuler et al [9,12] calculated the DOS of a charge in a RMF using the semiclassical approximation.…”

“…Moreover, in the case of RMF, the perturbative approach is also fundamentally problematic since one has to deal with the non-diagonal part of the Green function, which is not gauge invariant. In addition, the calculation of the Green function is plagued by infrared divergencies [7,8,9,10,11] that are due to the long-range nature of the correlations of the vector potential, even if the spatial correlations in the RMF are short-ranged. It has been suggested that these divergencies are due to the non-gauge-invariance of the Green function and therefore unphysical [9], although, recently, a physical interpretation has been proposed [11].…”