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arXiv:1912.10904 · Journal of Mathematical Physics

Holomorphic quantum Hall states in higher Landau levels

Nicolas Rougerie, Jakob Yngvason

Published Apr 1, 2020• 12 citations
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Abstract

Eigenstates of the planar magnetic Laplacian with a homogeneous magnetic field form degenerate energy bands, the Landau levels. We discuss the unitary correspondence between states in higher Landau levels and those in the lowest Landau level, where wave functions are holomorphic. We apply this correspondence to many-body systems; in particular, we represent effective Hamiltonians and particle densities in higher Landau levels by using corresponding quantities in the lowest Landau level.

DOI
10.1063/5.0004111

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•Aug 18, 2015•0 citations
Eigenstates of the planar magnetic Laplacian with a homogeneous magnetic field form degenerate energy bands, the Landau levels. We discuss the unitary correspondence between states in higher Landau levels and those in the lowest Landau level, where wave functions are holomorphic. We apply this correspondence to many-body systems; in particular, we represent effective Hamiltonians and particle densities in higher Landau levels by using corresponding quantities in the lowest Landau level.
We put forward possible wave functions for quantum Hall states in the lowest Landau level. These were deduced from the topological approach based on the relation between braid groups and the quantum statistics, as well as the commensurability condition unavoidable for collective states in magnetic fields. In this paper we demonstrate that the [Formula: see text]-field imposes restrictions on braid trajectories (i.e. elements of the full braid group). This results in the appearance of cyclotron subgroups, instead of the full braid group, for certain filling factors. The fermion representation of cyclotron subgroups defines transformations of wave functions in the quantum Hall regime. Hence, it sets quantum statistics (transformations of [Formula: see text] under exchanges of arguments), which is unavoidable for collective states (in compliance with the framework of Feynman's path integrals). Finally, the topological approach allows to define the hierarchy of fillings in the lowest Landau level, which agree with the hierarchy observed in quantum Hall devices (i.e. in transport measurements). The symmetry of a many-body wave function (i.e. quantum statistics) is always determined by a 1D unitary representation of the system's braid group. Using this topologically-originated property, we demonstrate that many-body wave functions for selected fillings of the lowest Landau level may not be purely antisymmetric. Only systems composed of fermions are investigated. Additionally, we present Monte Carlo calculations in a disc geometry, which remain in a nice agreement with predictions of exact diagonalizations (expected values of potential energy and pair distribution functions are presented). No boundary potential is assumed.
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Exact diagonalization of a two-dimensional electron gas in a strong magnetic field in the disk geometry shows that there exists a filling-factor range in the second Landau level where the states significantly differ from those in the lowest Landau level. We show that the difference arises because the interaction between composite fermions is not negligible in higher Landau levels, as indicated by a substantial mixing between composite-fermion Landau-like levels, or $\ensuremath{\Lambda}$ levels. We find that the exact ground state is well reproduced by composite-fermion theory with $\ensuremath{\Lambda}$-level mixing incorporated at the lowest level of approximation. Using the same variational approach in the spherical geometry we estimate the excitation gap at filling 1∕31∕31∕3 in the second Landau level (which corresponds to 7∕37∕37∕3 of experiment).
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 [Kapit and 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 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.
The Hamiltonian describing interacting two-dimensional electrons in a high magnetic field is diagonalized numerically for a small number of particles to obtain the low-lying excitation spectra. The results include estimates of energy gaps for values of $\ensuremath{\nu}$ (the lowest-Landau-level filling factor) equal to certain multiples of 15\frac{1}{5}51​, 17\frac{1}{7}71​, 19\frac{1}{9}91​, and 111\frac{1}{11}111​. These $\ensuremath{\nu}'\mathrm{s}$ are characterized by the existence of periodic rigid parent states which generate maximum phase space. The even-denominator cases are markedly different.
After more than three decades the fractional quantum Hall effect still poses challenges to contemporary physics. While many features are phenomenologically understood, a detailed comprehension of its microscopic origin still lacks. Here, we address this problem by showing that the Hamiltonian describing interacting electrons in a strong magnetic field, restricted to the lowest Landau level, can be mapped onto a one-dimensional lattice gas with repulsive interactions. The statistical mechanics of such models leads to interpret the sequence of Hall plateaux as a fractal phase diagram. We derive an expression for the ground states, forming Wigner crystals in the reciprocal lattice of the degenerate Landau states, and predict the form of the fractional plateaux landscape.