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Symmetry between quasielectrons and quasiholes for fractional quantum Hall models defined on lattices

Anne E. B. Nielsen, Ivan Glasser, Ivan Dario Jimenez Rodriguez

Published Sep 8, 2016• 0 citations
OpenAlex

Abstract

Quasielectrons in fractional quantum Hall systems are known to be much harder to describe theoretically than quasiholes. The problem is that one obtains a singularity in the wavefunction if one tries to naively construct the quasielectron as the inverse of the quasihole. Here we demonstrate that the same problem does not arise in lattice fractional quantum Hall models, so that quasielectrons can indeed be obtained as the inverse of quasiholes. Using this significantly simplified description, which can be applied quite generally, we compute braiding properties of quasielectrons and quasiholes and show that the charge distribution of quasielectrons is minus the charge distribution of quasiholes for different models in the disc and the torus geometry. We also derive few-body Hamiltonians, for which various states containing quasielectrons are ground states.

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Quasielectrons in fractional quantum Hall systems are known to be much harder to describe theoretically than quasiholes. The problem is that one obtains a singularity in the wavefunction if one tries to naively construct the quasielectron as the inverse of the quasihole. Here we demonstrate that the same problem does not arise in lattice fractional quantum Hall models, so that quasielectrons can indeed be obtained as the inverse of quasiholes. Using this significantly simplified description, which can be applied quite generally, we compute braiding properties of quasielectrons and quasiholes and show that the charge distribution of quasielectrons is minus the charge distribution of quasiholes for different models in the disc and the torus geometry. We also derive few-body Hamiltonians, for which various states containing quasielectrons are ground states.
From an experimental point of view, quasielectrons and quasiholes play very similar roles in the fractional quantum Hall effect. Nevertheless, the theoretical description of quasielectrons is known to be much harder than that of quasiholes. The problem is that one obtains a singularity in the wavefunction if one tries to naively construct the quasielectron as the inverse of the quasihole. Here, we demonstrate that the same problem does not arise in lattice fractional quantum Hall models. This result allows us to make detailed investigations of the properties of quasielectrons, including their braiding statistics and density distribution on lattices on the plane and on the torus. We show that some of the states considered have high overlap with certain fractional Chern insulator states. We also derive few-body Hamiltonians, for which various states containing quasielectrons are exact ground states.
From an experimental point of view, quasielectrons and quasiholes play very similar roles in the fractional quantum Hall effect. Nevertheless, the theoretical description of quasielectrons is known to be much harder than that of quasiholes. The problem is that one obtains a singularity in the wavefunction if one tries to naively construct the quasielectron as the inverse of the quasihole. Here, we demonstrate that the same problem does not arise in lattice fractional quantum Hall models. This result allows us to make detailed investigations of the properties of quasielectrons, including their braiding statistics and density distribution on lattices on the plane and on the torus. We show that some of the states considered have high overlap with certain fractional Chern insulator states. We also derive few-body Hamiltonians, for which various states containing quasielectrons are exact ground states.
Moore-Read states can be expressed as conformal blocks of the underlying rational conformal field theory, which provides a well explored description for the insertion of quasiholes. It is known, however, that quasielectrons are more difficult to describe in continuous systems, since the natural guess for how to construct them leads to a singularity. In this work, we show that the singularity problem does not arise for lattice Moore-Read states. This allows us to construct Moore-Read Pfaffian states on lattices for filling fraction 5/25/25/2 with both quasiholes and quasielectrons in a simple way. We investigate the density profile, charge, size, and braiding properties of the anyons by means of Monte Carlo simulations. Further we derive an exact few-body parent Hamiltonian for the states. Finally, we compare our results to the density profile, charge, and shape of anyons in the Kapit-Mueller model by means of exact diagonalization.
We present model wave functions for quasielectron (as opposed to quasihole) excitations of the unitary Z_{k} parafermion sequence (Laughlin, Moore-Read, or Read-Rezayi) of fractional quantum Hall states. We uniquely define these states through two generalized clustering conditions: they vanish when either a cluster of k+2 electrons is put together or when two clusters of k+1 electrons are formed at different positions. For Abelian fractional quantum Hall states (k=1), our construction reproduces the Jain quasielectron wave function and elucidates the difference between the Jain and Laughlin quasielectrons.
The quasiparticles in quantum Hall liquids carry fractional charge and obey fractional quantum statistics. Of particular recent interest are those with non-Abelian statistics, since their braiding properties could, in principle, be used for robust coding of quantum information. There is already a good theoretical understanding of quasiholes in both Abelian and non-Abelian quantum Hall states. Here we develop conformal field theory methods that allow for an equally precise description of quasielectrons and explicitly construct two- and four-quasielectron excitations of the non-Abelian Moore-Read state.
There has been significant interest in recent years in finding fractional quantum Hall physics in lattice models, but it is not always clear how these models connect to the corresponding models in continuum systems. Here we introduce a family of models that is able to interpolate between a recently proposed set of lattice models with Laughlin-like ground states constructed from conformal field theory and models with ground states that are practically the usual bosonic/fermionic Laughlin states in the continuum. Both the ground state and the Hamiltonian are known analytically, and we find that the Hamiltonian in the continuum limit does not coincide with the usual delta interaction Hamiltonian for the Laughlin states. We introduce quasiholes into the models and show analytically that their braiding properties are as expected if the quasiholes are screened. We demonstrate screening numerically for the 1/3 Laughlin model and find that the quasiholes are slightly smaller in the continuum than in the lattice. Finally, we compute the effective magnetic field felt by the quasiholes and show that it is close to uniform when approaching the continuum limit. The techniques presented here to interpolate between the lattice and the continuum can also be applied to other fractional quantum Hall states that are constructed from conformal field theory.