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arXiv:2004.12205 · Journal of Statistical Mechanics Theory and Experiment

Continuum limit of lattice quasielectron wavefunctions

Aniket Patra, Birgit Hillebrecht, Anne E. B. Nielsen

Published Aug 1, 2021• 2 citations
arXivHTMLPDF

Abstract

Abstract Trial states describing anyonic quasiholes in the Laughlin state were found early on, and it is therefore natural to expect that one should also be able to create anyonic quasielectrons. Nevertheless, the existing trial wavefunctions for quasielectrons show behaviors that are not compatible with the expected topological properties or their construction involves ad hoc elements. It was shown, however, that for lattice fractional quantum Hall systems, it is possible to find a relatively simple quasielectron wavefunction that has all the expected properties (2018 New J. Phys. 20 033029). This naturally poses the question: what happens to this wavefunction in the continuum limit? Here we demonstrate that, although one obtains a finite continuum wavefunction when the quasielectron is on top of a lattice site, such a limit of the lattice quasielectron does not exist in general. In particular, if the quasielectron is put anywhere else than on a lattice site, the lattice wavefunction diverges when the continuum limit is approached. The divergence can be removed by projecting the state on the lowest Landau level, but we find that the projected state does also not have the properties expected for anyonic quasielectrons. We hence conclude that the lattice quasielectron wavefunction does not solve the difficulty of finding trial states for anyonic quasielectrons in the continuum.

DOI
10.1088/1742-5468/ac0f63

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Abstract Trial states describing anyonic quasiholes in the Laughlin state were found early on, and it is therefore natural to expect that one should also be able to create anyonic quasielectrons. Nevertheless, the existing trial wavefunctions for quasielectrons show behaviors that are not compatible with the expected topological properties or their construction involves ad hoc elements. It was shown, however, that for lattice fractional quantum Hall systems, it is possible to find a relatively simple quasielectron wavefunction that has all the expected properties (2018 New J. Phys. 20 033029). This naturally poses the question: what happens to this wavefunction in the continuum limit? Here we demonstrate that, although one obtains a finite continuum wavefunction when the quasielectron is on top of a lattice site, such a limit of the lattice quasielectron does not exist in general. In particular, if the quasielectron is put anywhere else than on a lattice site, the lattice wavefunction diverges when the continuum limit is approached. The divergence can be removed by projecting the state on the lowest Landau level, but we find that the projected state does also not have the properties expected for anyonic quasielectrons. We hence conclude that the lattice quasielectron wavefunction does not solve the difficulty of finding trial states for anyonic quasielectrons in the continuum.
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.
New trial wave functions corresponding to half filling quantum Hall states are proposed. These wave functions are constructed by first pairing up the quasielectrons of the 1/3 Laughlin quantum Hall state, with the same relative angular momentum for each pair, and then making the paired quasielectrons condense into a 1/4 Laughlin state. The quasiparticle excitations of the proposed wave functions carry ±1/4\pm1/4±1/4 of electron charge, and obey Abelian fractional statistics. In the spherical geometry, the total flux quanta NϕN_ϕNϕ​ is shown to be related to the number of electrons NNN by Nϕ=2N−(5−q)N_ϕ = 2N-(5-q)Nϕ​=2N−(5−q) with qqq being the relative angular momentum between the quasielectrons in each pair which takes values of non-negative even integers. The overlaps are calculated between the proposed trial wave functions, including the ground state, quasiexciton states, and quasihole states, and the exact states of the finite size systems at Nϕ=2N−3N_ϕ = 2N-3Nϕ​=2N−3. The near unity overlaps are obtained in the lowest Landau level, while the moderate overlaps are obtained in the second Landau level. The relevance of the wave functions to the yet to be discovered fractional quantum Hall effect in the lowest Landau level, as well as the 5/2 quantum Hall effect is also discussed.
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.
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.
We discuss the link between the quasielectron wave functions proposed by Laughlin and by Jain and show both analytically and numerically that Laughlin's quasielectron is a nonlocal composite-fermion state. Composite-fermion states are typically discussed in terms of the composite-fermion Landau levels (also known as lambda levels). In standard composite-fermion quasielectron wave functions the excited lambda levels have subextensive occupation numbers. However, once the Laughlin's quasielectron is reformulated as a composite fermion, an overall logarithmic occupation of the first lambda level is made apparent, which includes orbitals that are localized at the boundary of the droplet. Even though the wave function proposed by Laughlin features a localized quasielectron with well-defined fractional charge, it exhibits some nontrivial boundary properties which motivate our interpretation of Laughlin's quasielectron as a nonlocal object. This has an important physical consequence: Laughlin's quasielectron fractionalizes an incorrect spin, deeply related to the anyonic braiding statistics. We conclude that Laughlin's quasielectron is not a good candidate for a quasielectron wave function.