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Hofstadter’s butterfly for a periodic array of quantum dots
Published in C Constanda, J Saranen, S Seikkala, Integral methods in science and engineering, 2020
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].
Boundary equation from a lattice model and modification of the Peierls equation
Published in Philosophical Magazine, 2022
The interaction range of solids is believed to be very short and the boundary layer, in which atoms interaction is distinct from that in interior, can be usually approximated as a monolayer. The boundary equation of a half-infinite lattice can be formally derived within lattice statics provided the interaction range is short enough. The boundary matrix as the kernal of the boundary equation can be straightforwardly calculated in wave vector space. Based on a useful decomposition of the boundary matrix, it is proved that the leading order correction to the boundary equation in elastic continuum theory comes from the boundary effects, which can be easily determined from the dynamics matrix. For a model of cubic lattice, the leading order correction of the boundary plane is presented explicitly. Surprisingly, in contrast to the naive guess [3, 6], the leading order correction in the isotropic approximation is not symmetric under rotation with the axis perpendicular to the boundary plane. If the leading order correction is divided into symmetric part and asymmetric part, the symmetric part is exactly the same as the continuum balance equation except a factor 1/2. For quasi one-dimensional problem, the boundary matrix is obtained analytically and the boundary equation with the discrete correction is given in the continuous approximation. As a application, the coefficients in front of the second differential term resulting from the lattice effects modification are successfully related to the elastic modulus of solids. The coefficient is for the edge dislocations and for the screw dislocation, where μ is the shear modulus and a is the period a in dislocation glide direction. For the narrow dislocation with the width comparable to the period, lattice effects modification is significant and should be taken into account.