Bloch oscillations of magnetic solitons as an example of dynamical localization of quasiparticles in a uniform external field (Review)

Abstract

The theory of the oscillatory motion of a band particle or particlelike excitation in a uniform field—the so-called Bloch oscillations—is reviewed. It is explained that this unusual motion is contingent on two circumstances: the time dependence of the motion of the particle under the influence of the external fields is governed by a classical equation of motion (dp/dt=F), and the energy spectrum of the particle is of a band nature, which presupposes a periodic dependence of the energy of the particle on its momentum (quasimomentum) ε=ε(p)=ε(p+p0), where p0, the period in p space, arises in a natural way in the description of the motion in a spatially periodic structure (lattice). Quasiclassical and quantum descriptions of the Bloch oscillations are given. Since a systematic exposition of the theory of this phenomenon has not been set forth in any monographs, the first part of this review gives a rather detailed presentation (with all the basic calculations) of the results on the oscillatory dynamics of an elementary excitation of a one-dimensional discrete chain, including the theory of the motion both in a uniform static field and in a uniform field with a harmonic time dependence. An interpretation is offered for the relationship of the frequency of quasiclassical Bloch oscillations and the equidistant spectrum of energy levels in the so-called Wannier–Stark ladder. An explanation of the physical nature of the phenomenon of dynamical localization of a band particle in a spatially uniform alternating field is given. It is shown that the basic results of such a dynamics carry over to the motion of a dynamical soliton of the discrete nonlinear Schrödinger equation. The second part of this review describes the Bloch oscillations of topological and dynamical magnetic solitons. It is shown that the phenomenological Landau–Lifshitz equations for the magnetization field in a magnetically ordered medium have surprising soliton solutions. The energy of a soliton is a periodic function of its momentum, even though its motion occurs in a continuous medium. The presence of this periodicity is sufficient to explain the Bloch oscillations of magnetic solitons. The quantum oscillatory dynamics of a soliton in a discrete spin chain is described. The review concludes with a discussion of the conditions for this oscillatory motion and the possibilities for its experimental observation.