Landau levels in lattices with long - range hopping
Author
Atakişi, Hakan
Oktel, M. Özgür
Date
2013Source Title
Physical Review A
Print ISSN
1050-2947
Publisher
American Physical Society
Volume
88
Issue
3
Pages
033612-1 - 033612-10
Language
English
Type
ArticleItem Usage Stats
143
views
views
97
downloads
downloads
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 expressionsChern numbers
Discretizations
Fractional quantum
General method
Hopping models
Landau levels
Periodic potentials
Quantum hall effect
Restoration
Wave functions