ABSTRACT
Beginning with the classical works of Azbel, Hofstadter, and Wannier (see [1]–3]), unusual spectral properties of two-dimensional periodic systems in a uniform magnetic field attract ever increasing attention. The main results here are concerned with theoretical explanations of the quantum Hall effect discovered by K. von Klitzing [4] (Nobel Prize, 1985). From the mathematical point of view this subject is of considerable interest because of its relations to a number of modern areas of mathematics: theory of characteristic classes, non-commutative geometry, operator extension theory, fractal geometry, etc. (see, for example, the review articles [5]–[7]). The most interesting properties of periodic systems with a magnetic field are conditioned by the presence of two natural geometric scales, namely, the magnetic length and the size of an elementary cell of the period lattice. The commensurability (or incommensurability) of the scales leads to such a pecularity of the systems as the transition from a band structure of the spectrum to a fractal one. This transition is described by the flux–energy diagram known as the “Hofstadter butterfly” (see [2] and [3]). Nevertheless, because the number of flux quanta through a unit cell of a crystal lattice is very small for experimentally accessible values of the magnetic field strength, no energy spectrum of the Hofstadter type is observable in usual Hall systems. However, artificial two-dimensional periodic-modulated systems (so-called periodic arrays of quantum dots) have recently been produced in which the above geometric scales are comparable and, consequently, an experimental observation of the Hofstadter butterfly has become possible [8].
