Quasielectrons as inverse quasiholes in lattice fractional quantum Hall models

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.

arXiv:1609.02389New Journal of PhysicsMar 12, 201817 citations

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

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

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.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.

Sep 8, 20160 citations

Quasielectrons in lattice Moore-Read models

Sourav Manna, Julia Wildeboer, Anne E. B. Nielsen

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 $5/2$ 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.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 $5/2$ 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.

arXiv:1810.12288Physical review. B./Physical review. BJan 28, 20197 citations

Continuum limit of lattice quasielectron wavefunctions

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

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.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.

arXiv:2004.12205Journal of Statistical Mechanics Theory and ExperimentAug 1, 20212 citations

Quasiparticles in the Quantum Hall Effect

Janik Kailasvuori

The fractional quantum Hall effect (FQHE), discovered in 1982 in a two-dimensional electron system, has generated a wealth of successful theory and new concepts in condensed matter physics, but is still not fully understood. The possibility of having nonabelian quasiparticle statistics has recently attracted attention on purely theoretical grounds but also because of its potential applications in topologically protected quantum computing. This thesis focuses on the quasiparticles using three different approaches. The first is an effective Chern-Simons theory description, where the noncommutativity imposed on the classical space variables captures the incompressibility. We propose a construction of the quasielectron and illustrate how many-body quantum effects are emulated by a classical noncommutative theory. The second approach involves a study of quantum Hall states on a torus where one of the periods is taken to be almost zero. Characteristic quantum Hall properties survive in this limit in which they become very simple to understand. We illustrate this by giving a simple counting argument for degeneracy 2n-1, pertinent to nonabelian statistics, in the presence of 2n quasiholes in the Moore-Read state and generalise this result to 2n-k quasiholes and k quasielectrons. In the third approach, we study the topological nature of the degeneracy 2n-1 by using a recently proposed analogy between the Moore-Read state and the two-dimensional spin-polarized p-wave BCS state. We study a version of this problem where one can use techniques developed in the context of high-Tc superconductors to turn the vortex background into an effective gauge field in a Dirac equation. Topological arguments in the form of index theory gives the degeneracy 2n-1 for 2n vortices.The fractional quantum Hall effect (FQHE), discovered in 1982 in a two-dimensional electron system, has generated a wealth of successful theory and new concepts in condensed matter physics, but is still not fully understood. The possibility of having nonabelian quasiparticle statistics has recently attracted attention on purely theoretical grounds but also because of its potential applications in topologically protected quantum computing. This thesis focuses on the quasiparticles using three different approaches. The first is an effective Chern-Simons theory description, where the noncommutativity imposed on the classical space variables captures the incompressibility. We propose a construction of the quasielectron and illustrate how many-body quantum effects are emulated by a classical noncommutative theory. The second approach involves a study of quantum Hall states on a torus where one of the periods is taken to be almost zero. Characteristic quantum Hall properties survive in this limit in which they become very simple to understand. We illustrate this by giving a simple counting argument for degeneracy 2n-1, pertinent to nonabelian statistics, in the presence of 2n quasiholes in the Moore-Read state and generalise this result to 2n-k quasiholes and k quasielectrons. In the third approach, we study the topological nature of the degeneracy 2n-1 by using a recently proposed analogy between the Moore-Read state and the two-dimensional spin-polarized p-wave BCS state. We study a version of this problem where one can use techniques developed in the context of high-Tc superconductors to turn the vortex background into an effective gauge field in a Dirac equation. Topological arguments in the form of index theory gives the degeneracy 2n-1 for 2n vortices.

Jan 1, 20061 citation

Characterization of quasiholes in two-component fractional quantum Hall states and fractional Chern insulators in |C| =2 flat bands

Błażej Jaworowski, Nicolas Regnault, Zhao Liu

We perform an exact-diagonalization study of quasihole excitations for the two-component Halperin (221) state in the lowest Landau level and for several $\ensuremath{\nu}=1/3$ bosonic fractional Chern insulators in topological flat bands with Chern number $|C|$ =2. Properties including the quasihole size, charge, and braiding statistics are evaluated. For the Halperin (221) model state, we observe isotropic quasiholes with a clear internal structure, and obtain the quasihole charge and statistics matching the theoretical values. Interestingly, we also extract the same quasihole size, charge, and braiding statistics for the continuum model states of $|C|$ =2 fractional Chern insulators, although the latter possess a ``color-entangled'' nature that does not exist in ordinary two-component Halperin states. We also consider two real lattice models with a band having $|C|$ =2. There we find that a quasihole can exhibit much stronger oscillations of the density profile, while having the same charge and statistics as those in the continuum models.We perform an exact-diagonalization study of quasihole excitations for the two-component Halperin (221) state in the lowest Landau level and for several $\ensuremath{\nu}=1/3$ bosonic fractional Chern insulators in topological flat bands with Chern number $|C|$=2. Properties including the quasihole size, charge, and braiding statistics are evaluated. For the Halperin (221) model state, we observe isotropic quasiholes with a clear internal structure, and obtain the quasihole charge and statistics matching the theoretical values. Interestingly, we also extract the same quasihole size, charge, and braiding statistics for the continuum model states of $|C|$=2 fractional Chern insulators, although the latter possess a ``color-entangled'' nature that does not exist in ordinary two-component Halperin states. We also consider two real lattice models with a band having $|C|$=2. There we find that a quasihole can exhibit much stronger oscillations of the density profile, while having the same charge and statistics as those in the continuum models.

arXiv:1810.03458Physical review. B./Physical review. BJan 23, 201924 citations

5/2

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.

|C|

=2. Properties including the quasihole size, charge, and braiding statistics are evaluated. For the Halperin (221) model state, we observe isotropic quasiholes with a clear internal structure, and obtain the quasihole charge and statistics matching the theoretical values. Interestingly, we also extract the same quasihole size, charge, and braiding statistics for the continuum model states of

|C|

=2 fractional Chern insulators, although the latter possess a ``color-entangled'' nature that does not exist in ordinary two-component Halperin states. We also consider two real lattice models with a band having

|C|

=2. There we find that a quasihole can exhibit much stronger oscillations of the density profile, while having the same charge and statistics as those in the continuum models.