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

arXiv:2512.04430

Edge Hamiltonian for Free Fermion Quantum Hall Models

Simon Du, Martin Fraas, Abi Gopal, Nathan Singh

math-phquant-phSubmitted Dec 4, 2025• Announced Dec 5, 2025• 0 citations
arXivHTMLPDF

Abstract

We investigate a proposal of Kitaev for a microscopic construction of a Hamiltonian intended to describe the edge dynamics of a quantum Hall system. We show that the construction works in the setting of translation-invariant free-fermion Hamiltonians. In this case, the resulting edge Hamiltonian exhibits only edge modes, and these modes have the correct chirality.

Comments
15 pages, 7 figures

Related papers

Free Fermion frustration-free models have zero Hall conductance

Sven Bachmann, Zhiqian Du, Martin Fraas

We give a new proof that translation-invariant free Fermion frustration-free models have zero Hall conductance. Our proof is analytic in nature and it uses the bulk-edge correspondence. The central argument, which holds beyond the translation-invariant situation, is that in frustration-free models, edge modes cannot merge into the ground state band. We give an example of a gapped frustration-free model that is nonetheless gapless on the half-plane.

arXiv:2610.21172•Oct 15, 2026•0 citations

Majorana edge magnetization in the Kitaev honeycomb model

Tomonari Mizoguchi, Tohru Koma

We propose an approach to detect the peculiarity of Majorana fermions at the edges of Kitaev magnets. As is well known, a pair of Majorana edge modes is realized when a single complex fermion splits into real and imaginary parts which are, respectively, localized at the left and right edges of a sample magnet. Reflecting both of this peculiarity of the Majorana fermions and the ground-state degeneracy caused by the existence of the Majorana edge zero modes, the spins at the edges of the sample magnet are expected to behave as a peculiar ``free'' spin which exhibits a unidirectional magnetization without any transverse magnetization when applied a sufficiently weak external magnetic field. For the Kitaev honeycomb model, we obtain the expression of the Majorana edge magnetization by relying on standard techniques to diagonalize a free fermion Hamiltonian. The magnetization profile thus obtained indeed shows the expected behavior. We also elucidate the relation between the Majorana edge flat band and the bulk winding number from a weak topological point of view.

arXiv:1811.10895•Physical review. B./Physical review. B•May 15, 2019•14 citations

Integrability of the ν=4/3 fractional quantum Hall edge states

Yichen Hu, Biao Lian

We investigate the homogeneous chiral edge theory of the filling ν=4/3 fractional quantum Hall state, which is parameterized by a Luttinger liquid velocity matrix and an electron tunneling amplitude (ignoring irrelevant terms). We identify two solvable cases: one case where the theory gives two free chiral boson modes, and the other case where the theory yields one free charge 2e3 chiral fermion and two free chiral Bogoliubov (Majorana) fermions. For generic parameters, the energy spectrum from our exact diagonalization shows Poisson level spacing statistics (LSS) in each conserved charge and momentum sector, indicating the existence of hidden conserved quantities and the possibility that the generic edge theory of the ν=4/3 fractional quantum Hall state is integrable. We further show that a global symmetry preserving irrelevant nonlinear kinetic term will lead to the transition of LSS from Poisson to Wigner-Dyson at high energies. This further supports the possibility that the model without irrelevant terms is integrable.

arXiv:2112.00754•Quantum•Jun 14, 2022•2 citations

Refermionized theory of the edge modes of a fractional quantum Hall cloud

Alberto Nardin, Iacopo Carusotto

Making use of refermionization techniques, we map the nonlinear chiral Luttinger liquid model of the edge modes of a spatially confined fractional quantum Hall cloud developed in our recent work [Nardin et al., Phys. Rev. A 107, 033320 (2023)] onto a one-dimensional system of massive and interacting chiral fermions, whose mass and interactions are set by the filling factor of the quantum Hall fluid and the shape of the external confining potential at the position of the edge. As an example of the predictive power of the refermionized theory, we report a detailed study of the dynamic structure factor and of the spectral function of a fractional quantum Hall cloud. Among other features, our refermionized theory provides a physical understanding of the effective decay of the edge excitations and of the universal power-law exponents at the thresholds of the dynamic structure factor. The quantitative accuracy of the refermionized theory is validated against a full two-dimensional calculation based on a combination of exact diagonalization and Monte Carlo sampling.

arXiv:2305.00291•Physical Review A•Nov 6, 2025•1 citation

Entanglement spectroscopy of chiral edge modes in the quantum Hall effect

Benoit Estienne, Jean-Marie Stéphan

We investigate the entanglement entropy in the integer quantum Hall effect in the presence of an edge, performing an exact calculation directly from the microscopic two-dimensional wave function. The edge contribution is shown to coincide exactly with that of a chiral Dirac fermion, and this correspondence holds for an arbitrary collection of intervals. In particular, for a single interval, the celebrated conformal formula is recovered with left and right central charges $c+\overline{c}=1$. Using Monte Carlo techniques, we establish that this behavior persists for strongly interacting systems such as Laughlin liquids. This illustrates how entanglement entropy is not only capable of detecting the presence of massless degrees of freedom, but also of pinpointing their position in real space, as well as elucidating their nature.

arXiv:1911.10125•Physical review. B./Physical review. B•Mar 19, 2020•31 citations

Chiral Luttinger liquids and a generalized Luttinger theorem in fractional quantum Hall edges via finite-entanglement scaling

Dániel Varjas, Michael P. Zaletel, Joel E. Moore

We use bosonic field theories and the infinite system density matrix renormalization group method to study infinite strips of fractional quantum Hall states starting from microscopic Hamiltonians. Finite-entanglement scaling allows us to accurately measure chiral central charge, edge-mode exponents, and momenta without finite-size errors. We analyze states in the first and second levels of the standard hierarchy and compare our results to predictions of the chiral Luttinger liquid theory. The results confirm the universality of scaling exponents in chiral edges and demonstrate that renormalization is subject to universal relations in the nonchiral case. We prove a generalized Luttinger theorem involving all singularities in the momentum-resolved density, which naturally arises when mapping Landau levels on a cylinder to a fermion chain and deepens our understanding of non-Fermi liquids in one dimension.

arXiv:1306.4976•Physical Review B•Oct 18, 2013•20 citations
We investigate a proposal of Kitaev for a microscopic construction of a Hamiltonian intended to describe the edge dynamics of a quantum Hall system. We show that the construction works in the setting of translation-invariant free-fermion Hamiltonians. In this case, the resulting edge Hamiltonian exhibits only edge modes, and these modes have the correct chirality.
We propose an approach to detect the peculiarity of Majorana fermions at the edges of Kitaev magnets. As is well known, a pair of Majorana edge modes is realized when a single complex fermion splits into real and imaginary parts which are, respectively, localized at the left and right edges of a sample magnet. Reflecting both of this peculiarity of the Majorana fermions and the ground-state degeneracy caused by the existence of the Majorana edge zero modes, the spins at the edges of the sample magnet are expected to behave as a peculiar ``free'' spin which exhibits a unidirectional magnetization without any transverse magnetization when applied a sufficiently weak external magnetic field. For the Kitaev honeycomb model, we obtain the expression of the Majorana edge magnetization by relying on standard techniques to diagonalize a free fermion Hamiltonian. The magnetization profile thus obtained indeed shows the expected behavior. We also elucidate the relation between the Majorana edge flat band and the bulk winding number from a weak topological point of view.
We investigate the homogeneous chiral edge theory of the filling ν=4/3 fractional quantum Hall state, which is parameterized by a Luttinger liquid velocity matrix and an electron tunneling amplitude (ignoring irrelevant terms). We identify two solvable cases: one case where the theory gives two free chiral boson modes, and the other case where the theory yields one free charge 2e3 chiral fermion and two free chiral Bogoliubov (Majorana) fermions. For generic parameters, the energy spectrum from our exact diagonalization shows Poisson level spacing statistics (LSS) in each conserved charge and momentum sector, indicating the existence of hidden conserved quantities and the possibility that the generic edge theory of the ν=4/3 fractional quantum Hall state is integrable. We further show that a global symmetry preserving irrelevant nonlinear kinetic term will lead to the transition of LSS from Poisson to Wigner-Dyson at high energies. This further supports the possibility that the model without irrelevant terms is integrable.
Making use of refermionization techniques, we map the nonlinear chiral Luttinger liquid model of the edge modes of a spatially confined fractional quantum Hall cloud developed in our recent work [Nardin et al., Phys. Rev. A 107, 033320 (2023)] onto a one-dimensional system of massive and interacting chiral fermions, whose mass and interactions are set by the filling factor of the quantum Hall fluid and the shape of the external confining potential at the position of the edge. As an example of the predictive power of the refermionized theory, we report a detailed study of the dynamic structure factor and of the spectral function of a fractional quantum Hall cloud. Among other features, our refermionized theory provides a physical understanding of the effective decay of the edge excitations and of the universal power-law exponents at the thresholds of the dynamic structure factor. The quantitative accuracy of the refermionized theory is validated against a full two-dimensional calculation based on a combination of exact diagonalization and Monte Carlo sampling.
We investigate the entanglement entropy in the integer quantum Hall effect in the presence of an edge, performing an exact calculation directly from the microscopic two-dimensional wave function. The edge contribution is shown to coincide exactly with that of a chiral Dirac fermion, and this correspondence holds for an arbitrary collection of intervals. In particular, for a single interval, the celebrated conformal formula is recovered with left and right central charges c+c‾=1c+\overline{c}=1c+c=1. Using Monte Carlo techniques, we establish that this behavior persists for strongly interacting systems such as Laughlin liquids. This illustrates how entanglement entropy is not only capable of detecting the presence of massless degrees of freedom, but also of pinpointing their position in real space, as well as elucidating their nature.
We use bosonic field theories and the infinite system density matrix renormalization group method to study infinite strips of fractional quantum Hall states starting from microscopic Hamiltonians. Finite-entanglement scaling allows us to accurately measure chiral central charge, edge-mode exponents, and momenta without finite-size errors. We analyze states in the first and second levels of the standard hierarchy and compare our results to predictions of the chiral Luttinger liquid theory. The results confirm the universality of scaling exponents in chiral edges and demonstrate that renormalization is subject to universal relations in the nonchiral case. We prove a generalized Luttinger theorem involving all singularities in the momentum-resolved density, which naturally arises when mapping Landau levels on a cylinder to a fermion chain and deepens our understanding of non-Fermi liquids in one dimension.