Uniform Laughlin gap and stability under bounded scalar disorder

Proves a uniform energy gap of at least 1/25 above the Laughlin state on the sphere; claims it survives weak disorder.

2 papers September 24, 2026 Explained by AI

Laughlin's 1983 wave function describes electrons in a strong magnetic field forming the fractional quantum Hall state at filling 1/3. A simple energy rule has this state as its unique lowest state, and physicists expected a fixed energy gap above it, however large the system. One paper shows the gap is at least 1/25 on large spheres, and Lean checks this. A second paper claims the gap and unique ground state survive weak random potentials.

First step

1983

Robert Laughlin, wave function for the fractional quantum Hall state at filling 1/3

Significance

Major

The Laughlin spectral-gap conjecture is a long-standing open question in the mathematics of the fractional quantum Hall effect.

Trust

Formal proof listed

Field

Mathematical physics

2 papers

The explanation

Start with just two electrons on a sphere with a magnetic monopole at its centre. Give the sphere Q = 3 flux quanta, units of magnetic flux. In the lowest energy band, called the lowest Landau level, each electron has 4 possible states. Two electrons, which must be antisymmetric, then have 6 joint states. Their pair spin, a measure of how they orbit each other, is either 2 or 0. Five states have pair spin 2, which is relative angular momentum one, the closest approach. One state has pair spin 0, so the electrons stay further apart.

Haldane's energy rule charges each pair energy one in relative angular momentum one, and zero otherwiseSource 1. So here 5 states cost 1 and one state costs 0. The free state is the Laughlin state for N = 2. Its gap, the step from the lowest energy to the next, is 1. In general the rule uses N electrons and Q = 3(N − 1) flux quanta. This is filling 1/3: about three flux quanta per electron. Laughlin's wave function describes the fractional quantum Hall state at that fillingSource 2.

The Laughlin state always has energy 0. The question is whether the gap stays above a fixed size as N grows. Haldane and Rezayi found numerical evidence, but noted it was not a proofSource 3. The obstacle is that a gap needs a lower bound over every competing many-electron stateSource 4. Exact gaps were known only for truncated versions on thin cylinders and thin toriSource 5.

The second paper settles the spherical gap conjectureSource 6. For all large enough systems, the gap above the Laughlin state is at least 1/25Source 7. The full result is stronger. It is the operator inequality H² ≥ γH on the whole space of states, for some γ above 1/25. The number used is 0.04616733319001, while 1/25 is 0.04. The size where the bound starts is not made explicitSource 8.

The first paper adds disorder. It applies a weak, bounded outside potential, like impurities in a sample. It claims a unique ground state and a uniform gap surviveSource 9. The allowed strength exists, but the paper does not compute or optimize itSource 10.

How sure is this? Lean, a computer proof checker, proves the unperturbed results. These include the 1/25 gap at flux 3(N − 1) for large N. They also include H² ≥ γH for every γ below 0.04616733319001 at large flux. Stability under the outside potential and uniqueness of the perturbed ground state are outside Lean. That part rests on one long paper, and no specialist review of it has been published.

What it doesn't show. This does not give an explicit system size where the bound starts, and Lean does not check the stability under disorder.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major if it holds. The 1/25 gap has a Lean proof; the stability under weak disorder rests on an unreviewed paper.

Words used

Fractional quantum Hall state: A strongly correlated state of electrons in two dimensions in a strong magnetic field.
Filling 1/3: One electron for about every three units of magnetic flux.
Lowest Landau level: The lowest energy band an electron can occupy in a strong magnetic field.
Laughlin state: Laughlin's explicit wave function for filling 1/3, where every pair of electrons keeps well apart.
V₁ interaction: Haldane's energy rule: each electron pair in its closest orbit, relative angular momentum one, costs energy one.
Spectral gap: The step from the lowest energy to the next energy.
Flux quantum: The basic unit of magnetic flux through a surface.

The original papers (2)

Uniform Stability of the Spherical Laughlin Gap (PDF, opens in a new tab)
October 5, 2026
A Fock-space inequality and the Laughlin spectral gap (PDF, opens in a new tab)
September 24, 2026 Formal proof listed

Lean notes

All quoted lines (10)

Haldane's spherical geometry and pseudopotentials identify a particularly simple parent Hamiltonian: every pair pays energy one in relative angular momentum one and zero in the other pair sectors.
A Fock-space inequality and the Laughlin spectral gap Back to text
Laughlin's wave function gave an explicit description of the fractional quantum Hall state at filling $1/3$ .
A Fock-space inequality and the Laughlin spectral gap Back to text
Haldane and Rezayi's finite-size study supplied numerical evidence and explicitly distinguished the expected hard-core gap from a formal proof.
A Fock-space inequality and the Laughlin spectral gap Back to text
Its exact Laughlin zero mode is known, but a uniform lower bound on the positive spectrum requires control of all competing many-particle states.
A Fock-space inequality and the Laughlin spectral gap Back to text
Rigorous uniform gaps were established for truncated pseudopotentials in thin-cylinder and thin-torus geometries by Nachtergaele, Warzel, and Young and Warzel and Young.
Uniform Stability of the Spherical Laughlin Gap Back to text
We prove this bound for the full interaction on the round sphere, resolving positively the spherical fermionic Laughlin spectral-gap conjecture.
A Fock-space inequality and the Laughlin spectral gap Back to text
With coefficient one for each pair projector, the gap above the Laughlin state at filling $1/3$ on the round sphere is at least $1/25$ for all sufficiently large systems.
A Fock-space inequality and the Laughlin spectral gap Back to text
The proof does not give an explicit flux threshold or the endpoint coefficient $\gamma_*$ on finite spheres.
A Fock-space inequality and the Laughlin spectral gap Back to text
We prove that the fermionic Laughlin $V_1$ Hamiltonian at filling $1/3$ on expanding round spheres retains a unique ground state and a uniform spectral gap under sufficiently weak bounded real scalar one-body potentials projected to the lowest Landau level.
Uniform Stability of the Spherical Laughlin Gap Back to text
We give existential constants; the proof does not optimize the disorder threshold.
Uniform Stability of the Spherical Laughlin Gap Back to text

Other sources

openai/math, Lean scope note for family 269 At flux $q=3(N-1)$ and all sufficiently large particle numbers $N$ , every antisymmetric state has $V_1$ energy at least $1/25$ times its squared distance from the Laughlin ground-state line.
openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

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