Tight-binding Electrons On The Two-dimensional Triangular And Kagome Lattices: Quantum Hall Effects And Hofstadter Butterflies Under Uniform And Staggered Modulated Magnetic Fields

The quantum Hall effect is one of the most spectacular phenomena in condensed matter physics. The integer quantum Hall effect (IQHE), discovered by K.v.Klitzing etc. in 1980 is observable in the systems of the two-dimensional electron gas, at low temperatures and in the presence of a strong perpendicular magnetic field. Later, the the phenomenology of the IQHE in theory was explained by R.B.Laughlin based on a gauge-invariance argument. Now, the quantized Hall effect was observed in various lattices. The electronic properties of 2D periodic lattices immersed in a uniform magnetic field are of special interest and have been hot topics of many physicists since D.R.Hofstadter found the energy spectrum of the butterflies' structures for a square lattice in 1976.The essay will discuss the tight-binding electronic peculiar properties on 2D triangular and kagome lattices under magnetic fields. Firstly, we first revisit the basic properties of the tight-binding electrons on triangular and kagome lattices. Next the electronic properties is by discussed under the a uniform-flux part withφand a staggered-flux part with strengthΔφ. For convenient analysis, according to lattices are topologically equivalent, added bonds between next-nearest-neighbor sites of the square lattice in only on direction get one new triangular lattice, and added sites between nearest-neighbor sites and linked the new sites with bonds of the square lattice in only one direction get another new kagome lattice. We adopt Landau gauge and the corresponding periodical boundary conditions (PBCs) for convenient calculation. Various properties of the Hall conductances and the Hofstadter butterflies of energy spectra are mainly studied by the essay. Whenφis fixed, variation ofΔφleads to quantum Hall transitions and the Chern numbers of Landau subbands are redistributed between neighboring pairs. The energy spectra for the 2D triangular lattice and kagome lattice with nonzeroΔφ's have similar fractal structures but quite different energy gaps compared with the original Hofstadter butterflies ofΔφ= 0. Moreover, the fan-like structure of Landau levels at the low magnetic field region is also modified appreciably byΔφ. The major achievements of the essay have been publ ished (J. Phys.:Condens. Matter, the first author).