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      Landau levels in lattices with long - range hopping

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      Author
      Atakişi, Hakan
      Oktel, M. Özgür
      Date
      2013
      Source Title
      Physical Review A
      Print ISSN
      1050-2947
      Publisher
      American Physical Society
      Volume
      88
      Issue
      3
      Pages
      033612-1 - 033612-10
      Language
      English
      Type
      Article
      Item Usage Stats
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      Abstract
      Landau levels (LLs) are broadened in the presence of a periodic potential, forming a barrier for accurate simulation of the fractional quantum Hall effect using cold atoms in optical lattices. Recently, it has been shown that the degeneracy of the lowest Landau level (LLL) can be restored in a tight-binding lattice if a particular form of long-range hopping is introduced. In this paper, we investigate three problems related to such quantum Hall parent Hamiltonians in lattices. First, we show that there are infinitely many long-range hopping models in which a massively degenerate manifold is formed by lattice discretizations of wave functions in the continuum LLL. We then give a general method to construct such models, which is applicable to not only the LLL but also higher LLs. We use this method to give an analytic expression for the hoppings that restores the LLL, and an integral expression for the next LL. We also consider whether the space spanned by discretized LL wave functions is as large as the space spanned by continuum wave functions, and we find the constraints on the magnetic field for this condition to be satisfied. Finally, using these constraints and the first Chern numbers, we identify the bands of the Hofstadter butterfly that correspond to continuum LLs.
      Keywords
      Analytic expressions
      Chern numbers
      Discretizations
      Fractional quantum
      General method
      Hopping models
      Landau levels
      Periodic potentials
      Quantum hall effect
      Restoration
      Wave functions
      Permalink
      http://hdl.handle.net/11693/20810
      Published Version (Please cite this version)
      http://dx.doi.org/10.1103/PhysRevA.88.033612
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