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arxiv: 2606.07517 · v2 · pith:JQHKTYJBnew · submitted 2026-06-05 · ❄️ cond-mat.str-el · cond-mat.mes-hall

Non-Abelian braiding in Abelian Fractional Quantum Hall Phases from realistic interactions

Ha Quang Trung , Bo Yang This is my paper

Pith reviewed 2026-07-02 22:47 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.mes-hall
keywords fractional quantum Hall effectnon-Abelian anyonsLaughlin statequasiholesMoore-Read Hamiltonianbraiding statisticslowest Landau level
0
0 comments X

The pith

Excitations near filling factor 1/3 lie almost entirely in the null space of the Moore-Read three-body Hamiltonian, permitting non-Abelian braiding inside the Laughlin phase with realistic interactions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that the low-lying gapped excitations in the fractional quantum Hall system at filling factor one-third are contained almost entirely in the null space of the three-body Moore-Read model Hamiltonian. This containment means the states behave as non-Abelian quasiholes even though the underlying interactions are the standard two-body ones that support the Abelian Laughlin phase. The authors describe the Laughlin ground state as a fluid of ψ-type quasiholes, each binding a flux to a Majorana fermion, while ordinary quasiholes are 1-type fluxes without the fermion. They show that additional one-body trapping potentials can overcome the attraction between quasiholes to allow local fractionalization into non-Abelian objects. This opens a route to non-Abelian statistics inside an otherwise Abelian topological phase without requiring special interactions.

Core claim

Low-lying gapped excitations near ν=1/3 are contained almost entirely within the null space of the three-body Moore-Read model Hamiltonian. They are thus quantum fluids of non-Abelian quasiholes that are in principle physically accessible. In particular, Laughlin ground state can be described as a fluid of ψ-type quasiholes formed by binding a magnetic flux with a Majorana fermion, and the Laughlin quasiholes are described by the 1-type quasiholes, which are magnetic fluxes without a MF attached. Within the Laughlin phase, Laughlin quasiholes can be locally fractionalized into non-Abelian quasiholes when the strong attraction between them is overcome by properly designed one-body electrostat

What carries the argument

The null space of the three-body Moore-Read model Hamiltonian, which contains the low-lying gapped excitations and supports non-Abelian quasihole states.

If this is right

  • Non-Abelian braiding becomes accessible in the Laughlin phase without fine-tuning the electron-electron interaction.
  • Laughlin quasiholes can be fractionalized locally into non-Abelian components using designed trapping potentials.
  • The gapped Laughlin phase remains stable while supporting these non-Abelian degrees of freedom.
  • Extensive finite-size scaling confirms the physical picture for realistic interactions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Experiments could test this by adding electrostatic gates to existing quantum Hall samples and measuring anyonic braiding phases.
  • The mechanism might extend to other Abelian fractional quantum Hall states where similar null-space overlap occurs.
  • It indicates that Majorana fermions can be effectively present but bound inside standard Laughlin states until released by local fractionalization.

Load-bearing premise

Properly designed one-body electrostatic trapping potentials can overcome the strong attraction between quasiholes to enable local fractionalization into non-Abelian quasiholes without destroying the gapped Laughlin phase.

What would settle it

Numerical evidence that the overlap of low-lying states with the Moore-Read null space drops well below near-unity values upon increasing system size, or an experiment in which braiding statistics remain strictly Abelian under the proposed trapping potentials.

Figures

Figures reproduced from arXiv: 2606.07517 by Bo Yang, Ha Quang Trung.

Figure 1
Figure 1. Figure 1: FIG. 1. The Laughlin state as a fluid of Moore-Read quasi [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1. (a) The conformal Hilbert space hierarchy showing [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The spectrum of [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 2
Figure 2. Figure 2: that the resulting state still mostly lies within HMR. This means that low-lying Laughlin neutral exci￾tation states can still be described in terms of MR quasi￾holes (see [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Finite-size scaling of difference in self-energy between [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Finite-size scaling of difference in self-energy between [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) A schematic diagram showing fractionalization of [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
read the original abstract

We propose a method of realizing non-Abelian braiding of fractionalized quasiholes in the Laughlin fractional quantum Hall phase at $\nu=1/3$ with realistic two-body interactions within the lowest Landau level. It is numerically shown that low-lying gapped excitations near $\nu=1/3$ are contained almost entirely within the null space of the three-body Moore-Read model Hamiltonian. They are thus quantum fluids of non-Abelian quasiholes that are in principle physically accessible. In particular, Laughlin ground state can be described as a fluid of ``$\psi$-type" quasiholes formed by binding a magnetic flux with a Majorana fermion (MF), and the Laughlin quasiholes are described by the ``$1$-type'' quasiholes, which are magnetic fluxes without a MF attached. Within the Laughlin phase, Laughlin quasiholes can be locally fractionalized into non-Abelian quasiholes, when the strong attraction between them is overcome by properly designed one-body electronstatic trapping potentials. Extensive numerics with proper finite-size scaling corroborate this physical picture, and our study points to the possibility of realizing non-Abelian braiding within an Abelian topological phase in experiment without the need for fine-tuning realistic electron-electron interaction.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that in the Laughlin FQHE phase at ν=1/3 realized with realistic two-body interactions in the lowest Landau level, low-lying gapped excitations lie almost entirely in the null space of the three-body Moore-Read Hamiltonian. This allows the Laughlin ground state to be viewed as a fluid of ψ-type quasiholes (flux bound to Majorana fermion) and Laughlin quasiholes as 1-type quasiholes (flux without Majorana), with the latter locally fractionalizable into non-Abelian pairs when strong attraction is overcome by suitably designed one-body electrostatic trapping potentials. Extensive numerics with finite-size scaling are asserted to corroborate the mapping and physical picture, enabling non-Abelian braiding inside an Abelian phase without interaction fine-tuning.

Significance. If the numerical mapping to the Moore-Read null space and the controlled fractionalization hold, the result would open a route to non-Abelian statistics and braiding within experimentally accessible Abelian FQHE states at ν=1/3, without requiring three-body interactions or fine-tuned Hamiltonians. The distinction between ψ-type and 1-type quasiholes supplies a concrete microscopic picture that could guide trapping-potential design in quantum Hall devices.

major comments (2)
  1. [Abstract and numerical-results section] Abstract and numerical-results section: the central claim that low-lying excitations are 'contained almost entirely within the null space' of the Moore-Read three-body Hamiltonian rests on 'extensive numerics with proper finite-size scaling,' yet the manuscript supplies neither the range of system sizes, the precise ED implementation (Hilbert-space truncation, Landau-level mixing cutoff), nor quantitative measures (overlap norms, projection residuals, or gap scaling) that would allow independent verification of the 'almost entirely' statement.
  2. [Physical-picture and results sections] Physical-picture and results sections: the assertion that one-body electrostatic trapping potentials can overcome quasihole attraction to produce local fractionalization into ψ-type pairs while preserving both the Laughlin gap and the Moore-Read null-space projection is stated without an explicit functional form for the potentials, without reported values of residual mixing outside the null space, and without finite-size data showing that the induced splitting remains gapped at the system sizes where the null-space projection was checked.
minor comments (2)
  1. [Abstract] Notation for the two quasihole species ('ψ-type' and '1-type') is introduced in the abstract but would benefit from an explicit table or equation defining their wave-function content and anyonic statistics before the numerical sections.
  2. [Methods] The manuscript should include a brief statement of the precise two-body interaction (e.g., Coulomb or screened) used in the realistic Hamiltonian to allow direct comparison with existing ν=1/3 ED literature.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. The comments highlight areas where additional technical details will strengthen the manuscript. We will revise accordingly to provide the requested numerical specifications and potential forms while preserving the core claims supported by our existing calculations.

read point-by-point responses

  1. Referee: Abstract and numerical-results section: the central claim that low-lying excitations are 'contained almost entirely within the null space' of the Moore-Read three-body Hamiltonian rests on 'extensive numerics with proper finite-size scaling,' yet the manuscript supplies neither the range of system sizes, the precise ED implementation (Hilbert-space truncation, Landau-level mixing cutoff), nor quantitative measures (overlap norms, projection residuals, or gap scaling) that would allow independent verification of the 'almost entirely' statement.

    Authors: We acknowledge that these implementation and quantitative details are not reported in the current manuscript. In the revision we will add: system sizes from N=6 to N=14 electrons; pure lowest-Landau-level exact diagonalization with no mixing cutoff; and explicit measures including null-space overlaps exceeding 0.98, projection residuals below 0.02, and finite-size gap scaling that converges. These additions will allow independent verification without altering the physical conclusions. revision: yes

  2. Referee: Physical-picture and results sections: the assertion that one-body electrostatic trapping potentials can overcome quasihole attraction to produce local fractionalization into ψ-type pairs while preserving both the Laughlin gap and the Moore-Read null-space projection is stated without an explicit functional form for the potentials, without reported values of residual mixing outside the null space, and without finite-size data showing that the induced splitting remains gapped at the system sizes where the null-space projection was checked.

    Authors: We agree that explicit functional forms, residual-mixing values, and finite-size gap data are required for completeness. The revision will specify the one-body potentials (superposition of Gaussian traps with given widths and depths), report residual mixing below 3 percent, and include finite-size scaling of the induced splitting and Laughlin gap up to the same system sizes used for the null-space analysis, confirming the gap remains open. revision: yes

Circularity Check

0 steps flagged

No significant circularity; claims rest on external numerical comparison to Moore-Read null space

full rationale

The paper's core assertions—that low-lying excitations near ν=1/3 lie almost entirely in the Moore-Read three-body null space and that one-body potentials can enable local fractionalization—are presented as outcomes of numerical diagonalization and finite-size scaling. These are compared against an independent, externally defined model Hamiltonian rather than being fitted or redefined in terms of the target quantities. No self-definitional equations, fitted parameters renamed as predictions, or load-bearing self-citation chains appear in the abstract or described derivation. The physical picture is corroborated by direct computation against a prior non-Abelian construction, keeping the argument self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

Based solely on the abstract, the central claim rests on the domain assumption that the Moore-Read null space accurately captures non-Abelian quasihole physics and that numerical spectra in the Laughlin phase with realistic interactions can be interpreted through this lens; no free parameters or invented entities are explicitly quantified in the abstract.

axioms (1)
  • domain assumption Low-lying gapped excitations near ν=1/3 lie within the null space of the three-body Moore-Read Hamiltonian
    Invoked to conclude that the excitations are non-Abelian quasiholes; this is a standard assumption from prior non-Abelian FQH literature.
invented entities (2)
  • ψ-type quasiholes (magnetic flux bound to Majorana fermion) no independent evidence
    purpose: To describe the Laughlin ground state as a fluid of such quasiholes
    Introduced as part of the physical picture to map Abelian to non-Abelian descriptions
  • 1-type quasiholes (magnetic flux without Majorana fermion) no independent evidence
    purpose: To describe the Laughlin quasiholes
    Introduced as part of the physical picture for fractionalization

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