J. - N. Fuchs, Mosseri, R., and Vidal, J., “Landau levels in quasicrystals”, PHYSICAL REVIEW B, vol. 98, p. 165427, 2018.Abstract
Two-dimensional tight-binding models for quasicrystals made of plaquettes with commensurate areas are considered. Their energy spectrum is computed as a function of an applied perpendicular magnetic field. Landau levels are found to emerge near band edges in the zero-field limit. Their existence is related to an effective zero-field dispersion relation valid in the continuum limit. For quasicrystals studied here, an underlying periodic crystal exists and provides a natural interpretation to this dispersion relation. In addition to the slope (effective mass) of Landau levels, we also study their width as a function of the magnetic flux and identify two fundamental broadening mechanisms: (i) tunneling between closed cyclotron orbits and (ii) individual energy displacement of states within a Landau level. Interestingly, the typical broadening of the Landau levels is found to behave algebraically with the magnetic field with a nonuniversal exponent.
J. - N. Fuchs, Piechon, F., and Montambaux, G., “Landau levels, response functions and magnetic oscillations from a generalized Onsager relation”, SCIPOST PHYSICS, vol. 4, p. 024, 2018.Abstract
A generalized semiclassical quantization condition for cyclotron orbits was recently proposed by Gao and Niu {[}1], that goes beyond the Onsager relation {[}2]. In addition to the integrated density of states, it formally involves magnetic response functions of all orders in the magnetic field. In particular, up to second order, it requires the knowledge of the spontaneous magnetization and the magnetic susceptibility, as was early anticipated by Roth {[}3]. We study three applications of this relation focusing on two-dimensional electrons. First, we obtain magnetic response functions from Landau levels. Second we obtain Landau levels from response functions. Third we study magnetic oscillations in metals and propose a proper way to analyze Landau plots (i.e. the oscillation index n as a function of the inverse magnetic field 1 = B) in order to extract quantities such as a zero-field phase-shift. Whereas the frequency of 1 = B-oscillations depends on the zero-field energy spectrum, the zero-field phase-shift depends on the geometry of the cell-periodic Bloch states via two contributions: the Berry phase and the average orbital magnetic moment on the Fermi surface. We also quantify deviations from linearity in Landau plots (i.e. aperiodic magnetic oscillations), as recently measured in surface states of three-dimensional topological insulators and emphasized by Wright and McKenzie {[}4].
G. Montambaux, Lim, L. - K., Fuchs, J. - N., and Piechon, F., “Winding Vector: How to Annihilate Two Dirac Points with the Same Charge”, PHYSICAL REVIEW LETTERS, vol. 121, p. 256402, 2018.Abstract
The merging or emergence of a pair of Dirac points may be classified according to whether the winding numbers which characterize them are opposite (+- scenario) or identical (++ cenario). From the touching point between two parabolic bands (one of them can be flat), two Dirac points with the same winding number emerge under appropriate distortion (interaction, etc.), following the ++ scenario. Under further distortion, these Dirac points merge following the +- scenario, that is corresponding to opposite winding numbers. This apparent contradiction is solved by the fact that the winding number is actually defined around a unit vector on the Bloch sphere and that this vector rotates during the motion of the Dirac points. This is shown here within the simplest two-band lattice model (Mielke) exhibiting a flat band. We argue on several examples that the evolution between the two scenarios is general.