Energy spectrum for two-dimensional potentials in very high magnetic fields

Abstract

A method, analogous to supersymmetry transformation in quantum mechanics, is developed for a particle in the lowest Landau level moving in an arbitrary potential. The method is applied to two-dimensional potentials formed by Dirac ␦ scattering centers. In the periodic case, the problem is solved exactly for rational values of the magnetic flux ͑in units of flux quantum͒ per unit cell. The spectrum is found to be self-similar, resembling the Hofstadter butterfly ͓Phys. Rev. B 14, 2239 ͑1976͔͒. ͓S0163-1829͑97͒06436-9͔In recent years, the energy spectrum of two-dimensional electron systems have attracted great interest, because of the relevance of the problem to the magnetotransport properties 1 and in particular to the quantum Hall effect. 2 It is believed that the physics of the integer quantum Hall effect is governed by the interaction of electrons with a disordered potential, which leads to a localization of the eigenfunctions. Observed conductance steps can be explained by a sequence of localization-delocalization transitions. Quantization of the Hall conductance due to periodic potentials has also been studied. 3,4 In this case, the presence of steps is explained by the gaps in the energy spectrum.There are two opposite approaches to the problem of an electron moving in a periodic potential: the tight-binding and the nearly-free-electron methods. In the first approach, the magnetic field is introduced via Peierls substitution, where the matrix elements are multiplied by exp͓(iq/បc) ͐A•dl͔. 5 On the other hand, in the case of nearly free electrons, the Landau-level structure is essential, and the lattice potential is introduced via intra-and inter-Landau-level scattering matrix elements. The duality between the position and the momentum in quantum mechanics leads to similarities between the two methods. In the presence of a magnetic field, the secular equations for the two limits, the tight-binding and the nearlyfree-electron approach, with certain approximations, are identical. 6 The characteristic feature of the problem is that the secular determinant for the limit of infinite crystals can be reduced to a finite determinant, when the magnetic flux per unit cell is a rational number in units of the flux quantum. The tight-binding case was studied by Azbel' 7 and Hofstadter, 8 who showed that the system has a complicated self-similar spectrum.In this work, an approach is developed for a particle in the lowest Landau level moving in an arbitrary potential. The method is used to obtain the energy spectrum in the presence of Dirac ␦ potentials. The difficulty with the Dirac ␦ potential is that, even if the inter-Landau-level couplings are small and can be neglected, there is a strong intra-Landau-level mixing. The formalism developed in this study leads to an eigenvalue problem where the coupling between distant sites become negligibly small. Moreover, since the problem is formulated in real space, the distribution of Dirac ␦ potentials can be arbitrary. In spite of the sharpness of the potential, the assumption t...

Self-similar occurrence of massless Dirac particles in graphene under a magnetic field

“…(8) and (9) with the appropriate momenta given by Eqs. (15) and (16). At this point, it is illuminating to obtain the locations of the zero-energy momenta in the magnetic Brillouin zone.…”

“…[6][7][8][9] In addition to numerical studies solving Harper's equation, there have been extensive efforts to obtain analytic solutions. [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] The reason for such efforts is multifaceted. For one, many researchers have been curious about the very origin of the self-similar fractal structure seen in the Hofstadter butterfly and tried to make a connection to other known systems exhibiting similar fractal structures.…”

Self-Similar Optical Absorption Spectra in High Magnetic Fields

“…The main challenges in a correct description of this situation lie in the fact that (i) the magnetic field destroys the original periodicity of the problem and (ii) the vector potential associated with a constant magnetic field A k x is an unbound function that hampers the use of any kind of periodic boundary conditions. Several approaches have been developed to deal with this problem nonperturbatively [3][4][5][6]. So far, most theoretical studies focused on the electronic energy spectrum [4][5][6] and the Hall conductance [4] within one-band or many-band situations which both reflect the fractal eigenvalue spectrum of the Hofstadter butterfly.…”

“…Several approaches have been developed to deal with this problem nonperturbatively [3][4][5][6]. So far, most theoretical studies focused on the electronic energy spectrum [4][5][6] and the Hall conductance [4] within one-band or many-band situations which both reflect the fractal eigenvalue spectrum of the Hofstadter butterfly. It has not been clarified so far, however, whether the optical absorption spectrum is generally fractal as well or whether the robustness of Kohn's theorem [7] together with optical selection rules will mask the self-similar eigenvalue spectrum.…”

“…Several approaches have been developed to deal with this problem nonperturbatively [3][4][5][6]. So far, most theoretical studies focused on the electronic energy spectrum [4][5][6] and the Hall conductance [4] within one-band or many-band situations phys. stat.…”