Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Dirac Fermions on a 2D lattice in the presence of defects : with possible applications to quantum computation

William Conan Weeks

Published Jan 1, 2011• 0 citations
Publisher

Abstract

Dirac Fermions on 2D lattice are studied in the presence of a single flux tube, a pair of flux tubes and an edge dislocation/anti-dislocation pair. Numerical studies have been carried out using exact diagonalization for each case, and the results indicate that the model supports fractionally charged particles with anyonic statistics. An analytical calculation is also provided, predicting a zero energy bound state in the continuum for Dirac Fermions in the presence of a single flux tube carrying one-half flux quantum, as well as a brief discussion of anyonic quasiparticles and their possible application to quantum computation.

DOI
10.14288/1.0084932

Related papers

The Quantum Hall Problem In Lattices

Eliot Kapit

This thesis represents a body of work investigating the physics of strongly interacting quantum particles, confined to a two-dimensional plane in a nontrivial vector potential (such as a transverse magnetic field), commonly referred to as quantum Hall physics. The result which anchors these studies is my discovery, reported in 2010 (PRL 105, 215303), of a lattice model in which there is a degenerate manifold of single-particle states, with wavefunctions matching those of the lowest Landau level of continuum particles, providing a bridge between continuum and lattice physics. Within this model anyonic states are robust and thus amenable to observation. Building on this result, I numerically demonstrate the braiding of anyons in many-body quantum Hall states of bosons, confirming their anyonic statistics. I also study the equation of state of quantum Hall bosons for various flux densities and choices of hopping parameters, with the goal of quantifying the effects of finite temperature and examining the feasibility of observing these states in a system of cold atoms. I then propose a new architecture for superconducting qubits, where flux states of circulating current combined with superconducting transformers and tuned voltage offsets mimic the physics of charged particles in a magnetic field, allowing boson quantum Hall physics to be studied in an environment free of charge noise. Finally, I review the work presented here and speculate on its possible applications to topological quantum computing.

•Aug 20, 2012•0 citations

Lattice Defects in the Kitaev Honeycomb Model

John Brennan, Jiří Vala

The Kitaev honeycomb lattice system is an important model of topological materials whose phase diagram exhibits both abelian and non-abelian topological phases. The latter, a so-called Ising phase, is related to topological superconductors. Its quasiparticle excitations, which are formed by Majorana fermions attached to vortices, show non-abelian fractional statistics and are known as Ising anyons. We investigate dislocation defects in the Ising phase of the Kitaev honeycomb model. After introducing them to the system, we accordingly generalize our solution of this model to the situation with the defects. The important part of this effort is developing an appropriate Jordan-Wigner fermionization procedure. It is expected that the presence of defects manifests itself by the formation of fermionic zero-energy modes around the defect end points. We numerically confirm this expectation and further investigate properties of these modes. The computational potential of our technique is demonstrated for both diagonalization and dynamical simulations. The latter focuses on the process of fusion of the vortex zero-energy modes with the Majorana fermions attached to the defect. This process simulates fusion of non-abelian Ising anyons.

The Journal of Physical Chemistry A•Feb 17, 2016•13 citations

Disclinations, Dislocations, and Emanant Flux at Dirac Criticality

Maissam Barkeshli, Christopher Fechisin, Zohar Komargodski +1 more

What happens when fermions hop on a lattice with crystalline defects? The answer depends on topological quantum numbers that specify the action of lattice rotations and translations in the low energy theory. One can understand the topological quantum numbers as a twist of continuum gauge fields in terms of crystalline gauge fields. We find that disclinations and dislocations—defects of crystalline symmetries—generally lead to a certain “emanant” quantized magnetic flux in the continuum. To demonstrate these facts, we study in detail tight-binding models whose low-energy descriptions are ( 2 + 1 ) D Dirac cones. Our map from lattice to continuum defects explains the crystalline topological response to disclinations and dislocations, and motivates the fermion crystalline equivalence principle used in the classification of crystalline topological phases. When the gap closes, the presence of emanant flux leads to pair creation from the vacuum with the particles and antiparticles swirling around the defect. We compute the associated currents and energy density using the tools of defect conformal field theory. There is a rich set of renormalization group fixed points, depending on how particles scatter from the defect. At half flux, there is a defect conformal manifold leading to a continuum of possible low-energy theories. We present extensive numerical evidence supporting the emanant magnetic flux at lattice defects, and we test our map between lattice and continuum defects in detail. We also point out a no-go result, which implies that a single ( 2 + 1 ) D Dirac cone in symmetry class AII is incompatible with a commuting C M rotational symmetry with ( C M ) M = + 1 .

arXiv:2501.13866•Physical Review X•Dec 8, 2025•2 citations

Magnetic Thomas-Fermi theory for 2D abelian anyons

Antoine Levitt, Douglas Lundholm, Nicolas Rougerie

Two-dimensional abelian anyons are, in the magnetic gauge picture, represented as fermions coupled to magnetic flux tubes. For the ground state of such a system in a trapping potential, we theoretically and numerically investigate a Hartree approximate model, obtained by restricting trial states to Slater determinants and introducing a self-consistent magnetic field, locally proportional to matter density. This leads to a fermionic variant of the Chern-Simons-Schrödinger system. We find that for dense systems, a semi-classical approximation yields qualitatively good results. Namely, we derive a density functional theory of magnetic Thomas-Fermi type, which correctly captures the trends of our numerical results. In particular, we explore the subtle dependence of the ground state with respect to the fraction of magnetic flux units attached to particles.

arXiv:2504.13481•SciPost Physics Core•Mar 30, 2026•1 citation

Dirac fermions with disorder in two dimensions: Exact results

K. Ziegler

Dirac fermions on a two-dimensional lattice with disorder are considered. The Dirac mass, which controls the gap between the two bands of the fermions, is subject to random fluctuations. Another type of disorder is discussed presented by a random vector potential. It is shown that the imaginary part of the one-particle Green's function can be written as the imaginary part of another Green's function, which has only poles on the lower half plane. Therefore, it is possible to perform a Cauchy integration for a Lorentzian distribution in analogy with the Lloyd model. The results are compared with calculations performed in the continuum limit based on renormalization group and bosonization methods. \textcopyright{} 1996 The American Physical Society.

arXiv:cond-mat/9508091•Physical review. B, Condensed matter•Apr 15, 1996•16 citations

Anyons in one dimension

K.-V. Pham, M. Gabay, Pascal Lederer

Particles possessing fractional statistics in two dimension have been dubbed anyons. We want to show how it is possible to extend this concept in the case of conformally invariant systems in one dimension, and reproduce the flux-tube attachment construction due to Wilczek. We will introduce a class of models describing 1D anyons, which has the great advantage over its 2D counterparts to be exactly solvable. The properties of these 1D anyons generalize those of a Luttinger liquid and yield a complete classification of 1D fixed points when time-reversal symmetry is broken.

Journal de Physique IV (Proceedings)•Mar 1, 2000•2 citations
Dirac Fermions on 2D lattice are studied in the presence of a single flux tube, a pair of flux tubes and an edge dislocation/anti-dislocation pair. Numerical studies have been carried out using exact diagonalization for each case, and the results indicate that the model supports fractionally charged particles with anyonic statistics. An analytical calculation is also provided, predicting a zero energy bound state in the continuum for Dirac Fermions in the presence of a single flux tube carrying one-half flux quantum, as well as a brief discussion of anyonic quasiparticles and their possible application to quantum computation.
This thesis represents a body of work investigating the physics of strongly interacting quantum particles, confined to a two-dimensional plane in a nontrivial vector potential (such as a transverse magnetic field), commonly referred to as quantum Hall physics. The result which anchors these studies is my discovery, reported in 2010 (PRL 105, 215303), of a lattice model in which there is a degenerate manifold of single-particle states, with wavefunctions matching those of the lowest Landau level of continuum particles, providing a bridge between continuum and lattice physics. Within this model anyonic states are robust and thus amenable to observation. Building on this result, I numerically demonstrate the braiding of anyons in many-body quantum Hall states of bosons, confirming their anyonic statistics. I also study the equation of state of quantum Hall bosons for various flux densities and choices of hopping parameters, with the goal of quantifying the effects of finite temperature and examining the feasibility of observing these states in a system of cold atoms. I then propose a new architecture for superconducting qubits, where flux states of circulating current combined with superconducting transformers and tuned voltage offsets mimic the physics of charged particles in a magnetic field, allowing boson quantum Hall physics to be studied in an environment free of charge noise. Finally, I review the work presented here and speculate on its possible applications to topological quantum computing.
The Kitaev honeycomb lattice system is an important model of topological materials whose phase diagram exhibits both abelian and non-abelian topological phases. The latter, a so-called Ising phase, is related to topological superconductors. Its quasiparticle excitations, which are formed by Majorana fermions attached to vortices, show non-abelian fractional statistics and are known as Ising anyons. We investigate dislocation defects in the Ising phase of the Kitaev honeycomb model. After introducing them to the system, we accordingly generalize our solution of this model to the situation with the defects. The important part of this effort is developing an appropriate Jordan-Wigner fermionization procedure. It is expected that the presence of defects manifests itself by the formation of fermionic zero-energy modes around the defect end points. We numerically confirm this expectation and further investigate properties of these modes. The computational potential of our technique is demonstrated for both diagonalization and dynamical simulations. The latter focuses on the process of fusion of the vortex zero-energy modes with the Majorana fermions attached to the defect. This process simulates fusion of non-abelian Ising anyons.
What happens when fermions hop on a lattice with crystalline defects? The answer depends on topological quantum numbers that specify the action of lattice rotations and translations in the low energy theory. One can understand the topological quantum numbers as a twist of continuum gauge fields in terms of crystalline gauge fields. We find that disclinations and dislocations—defects of crystalline symmetries—generally lead to a certain “emanant” quantized magnetic flux in the continuum. To demonstrate these facts, we study in detail tight-binding models whose low-energy descriptions are ( 2 + 1 ) D Dirac cones. Our map from lattice to continuum defects explains the crystalline topological response to disclinations and dislocations, and motivates the fermion crystalline equivalence principle used in the classification of crystalline topological phases. When the gap closes, the presence of emanant flux leads to pair creation from the vacuum with the particles and antiparticles swirling around the defect. We compute the associated currents and energy density using the tools of defect conformal field theory. There is a rich set of renormalization group fixed points, depending on how particles scatter from the defect. At half flux, there is a defect conformal manifold leading to a continuum of possible low-energy theories. We present extensive numerical evidence supporting the emanant magnetic flux at lattice defects, and we test our map between lattice and continuum defects in detail. We also point out a no-go result, which implies that a single ( 2 + 1 ) D Dirac cone in symmetry class AII is incompatible with a commuting C M rotational symmetry with ( C M ) M = + 1 .
Two-dimensional abelian anyons are, in the magnetic gauge picture, represented as fermions coupled to magnetic flux tubes. For the ground state of such a system in a trapping potential, we theoretically and numerically investigate a Hartree approximate model, obtained by restricting trial states to Slater determinants and introducing a self-consistent magnetic field, locally proportional to matter density. This leads to a fermionic variant of the Chern-Simons-Schrödinger system. We find that for dense systems, a semi-classical approximation yields qualitatively good results. Namely, we derive a density functional theory of magnetic Thomas-Fermi type, which correctly captures the trends of our numerical results. In particular, we explore the subtle dependence of the ground state with respect to the fraction of magnetic flux units attached to particles.
Dirac fermions on a two-dimensional lattice with disorder are considered. The Dirac mass, which controls the gap between the two bands of the fermions, is subject to random fluctuations. Another type of disorder is discussed presented by a random vector potential. It is shown that the imaginary part of the one-particle Green's function can be written as the imaginary part of another Green's function, which has only poles on the lower half plane. Therefore, it is possible to perform a Cauchy integration for a Lorentzian distribution in analogy with the Lloyd model. The results are compared with calculations performed in the continuum limit based on renormalization group and bosonization methods. \textcopyright{} 1996 The American Physical Society.
Particles possessing fractional statistics in two dimension have been dubbed anyons. We want to show how it is possible to extend this concept in the case of conformally invariant systems in one dimension, and reproduce the flux-tube attachment construction due to Wilczek. We will introduce a class of models describing 1D anyons, which has the great advantage over its 2D counterparts to be exactly solvable. The properties of these 1D anyons generalize those of a Luttinger liquid and yield a complete classification of 1D fixed points when time-reversal symmetry is broken.