Landau Levels in Lattices with Long Range Hopping

Abstract

In the presence of a periodic potential Landau levels (LLs) are broadened, forming a barrier for accurate simulation of 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 [E. Kapit and E. J. Mueller, Phys. Rev. Lett. 105, 215303 (2010)]. 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 wavefunctions 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 restore the LLL, and an integral expression for the next LL. We also consider whether the space spanned by discretized LL wavefunctions is as large as the space spanned by continuum wavefunctions and find the constraints on magnetic field for this condition to be satisfied. Finally, using these constraints and first Chern numbers, we identify the bands of the Hofstadter butterfly that correspond to continuum LLs.

26 pages, 12 figures

References in corpus (3)

• Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms
• Time-reversal symmetry breaking in circuit-QED based photon lattices
• Composite Fermion Theory for Bosonic Atoms in Optical Lattices

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• Building Flat-Band Lattice Models from Gram Matrices
• Exact Topological Flat Bands from Continuum Landau Levels
• Structure of the nearly-degenerate manifold of lattice quasiholes on a torus
• Invisible flat bands on a topological chiral edge
• Lattice oscillator model on noncommutative space: eigenvalues problem for the perturbation theory
• Exact Parent Hamiltonians for All Landau Level States in a Half-flux Lattice