Uniform Stability of the Spherical Laughlin Gap

OpenAI

Published Oct 6, 2026• 0 citations

Abstract

We prove that the fermionic Laughlin V1 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. With coefficient one for each pair projector and Laughlin flux q=3(N-1), the gap and perturbation threshold are uniform for all sufficiently large particle numbers and all potential profiles of supremum norm at most one. The result uses the uniform unperturbed Fock-space gap and gives existential constants.

DOI: 10.83295/rvyv-r42z

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Uniform Stability of the Spherical Laughlin Gap

OpenAI

Published Oct 6, 2026• 0 citations

Publisher

Abstract

DOI: 10.83295/rvyv-r42z

Related papers

A Fock-space inequality and the Laughlin spectral gap

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Oct 6, 20260 citations

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Mikael Fremling, Jørgen Fulsebakke, Niall Moran +1 more

We develop a method to efficiently calculate trial wave functions for quantum Hall systems which involve projection onto the lowest Landau level. The method essentially replaces the lowest Landau level projection by projection onto the $M$ lowest eigenstates of a suitably chosen Hamiltonian acting within the lowest Landau level. The resulting ``energy projection'' is a controlled approximation to the exact lowest Landau level projection which improves with increasing $M$ . It allows us to study the projected trial wave functions for system sizes close to the maximal sizes that can be reached by exact diagonalization and can be straightforwardly applied in any geometry. As a first application and test case, we study a class of trial wave functions first proposed by Girvin and Jach [Girvin and Jach, Phys. Rev. B 29, 5617 (1984)], which are modifications of the Laughlin states involving a single real parameter. While these modified Laughlin states probably represent the same universality class exemplified by the Laughlin wave functions, we show by extensive numerical work for systems on the sphere and torus that they provide a significant improvement of the variational energy, overlap with the exact wave function and properties of the entanglement spectrum.We develop a method to efficiently calculate trial wave functions for quantum Hall systems which involve projection onto the lowest Landau level. The method essentially replaces the lowest Landau level projection by projection onto the $M$ lowest eigenstates of a suitably chosen Hamiltonian acting within the lowest Landau level. The resulting ``energy projection'' is a controlled approximation to the exact lowest Landau level projection which improves with increasing $M$. It allows us to study the projected trial wave functions for system sizes close to the maximal sizes that can be reached by exact diagonalization and can be straightforwardly applied in any geometry. As a first application and test case, we study a class of trial wave functions first proposed by Girvin and Jach [Girvin and Jach, Phys. Rev. B 29, 5617 (1984)], which are modifications of the Laughlin states involving a single real parameter. While these modified Laughlin states probably represent the same universality class exemplified by the Laughlin wave functions, we show by extensive numerical work for systems on the sphere and torus that they provide a significant improvement of the variational energy, overlap with the exact wave function and properties of the entanglement spectrum.

arXiv:1601.06736Physical review. B./Physical review. BJun 24, 201615 citations

Φ_h

be a free fermionic interaction, possibly including pairing terms, whose infinite-volume Majorana coefficient operator

h

has no spectrum in

(-g,g)

, and let

\mathcal V

be an even self-adjoint many-body interaction. For sufficiently small

|λ|

, we construct pure ground states

ω_λ

of the interacting bulk system

Φ_λ=Φ_h+λ\mathcal V

and quasi-local automorphisms

β_{0,λ}

such that

ω_λ=ω_0\circβ_{0,λ}

, where

ω_0

is the free ground state of

Φ_h

. Each

ω_λ

has a GNS Hamiltonian with gap at least

g-C_*|λ| \|\mathcal V\|_*

, where

\|\cdot\|_*

is the relevant interaction norm. Our result allows polynomial decay of both the free coefficients and the many-body interaction. For two-site interactions on a uniformly discrete point set

Γ\subset\mathbb{R}^d

, a sufficient condition is decay as

(1+|x-y|)^{-\varpi}

with

\varpi>2d

, for both the hopping and pairing coefficients and the interaction terms. The spectral-flow generator preserves the spatial exponent of the polynomial interaction norm, and its automorphisms satisfy locality bounds at that exponent. If

h

and

\mathcal V

have stretched-exponential decay, so does the spectral-flow generator. The proof constructs the spectral flow and establishes the gap through an excitation-number comparison, using polynomial propagation bounds for evolved interactions proved in a companion paper.

A recent preprint by OpenAI resolved the spherical fermionic Laughlin spectral-gap conjecture for the full

V_1

interaction by a Fock-space inequality:

H_Q^2\ge γH_Q

on the lowest-Landau-level Fock space for all large flux

Q

and every

γ

below

0.0461673\ldots

, so that the gap above the Laughlin state at filling

1/3

is at least

1/25

. That proof, with its original constant, has also been formalized in Lean. We show that the same argument supports a much larger constant. Two sharper estimates, a product-twirl bound for the three-body tail and a staircase price for the four-body allowance, raise the constant above

0.0601

without changing the original certificate. Treating the auxiliary operators as the variables of a semidefinite program (SDP) and enlarging the finite set of orbital modes on which they act then gives

H_Q^2\geγH_Q

for every

γ<γ_*=0.2190037845063196\ldots

, certified by exact rational arithmetic. Consequently the gap above the Laughlin state is at least

1/5

for all sufficiently large spheres, and the untruncated planar

V_1

Hamiltonian has positive spectrum bounded below by

γ_*

, uniformly in particle number and angular momentum. A heuristic extrapolation of the SDP optima suggests that auxiliary operators of the present form can give at most about

0.222

, while exact diagonalization of finite systems places the true gap near

0.4

. A larger family of auxiliary operators could push the bound further towards the true gap, and the SDP formulation makes such families easy to test. The proof of the improvement and the SDP search were completed by the AI model Claude Opus 5.5 (Anthropic) under the author's direction.

We develop a bootstrap approach to deriving lower bounds on the spectral gap of the first Haldane pseudopotential Hamiltonian on the sphere with

N_Φ

flux quanta for arbitrary

N_Φ

, including in the thermodynamic limit

N_Φ\to \infty

. This method can in principle be generalized to pseudopotentials describing other model wavefunctions such as the Moore--Read state or Read--Rezayi sequence. Our approach is based on Knabe's observation that if

H

is a Hamiltonian with ground state energy zero then the spectral gap is at least

ε> 0

if and only if

H^2 \succeq εH

. To show this inequality, we split

H^2

into two parts, one that can be analytically bounded and another that can be bounded using a finite semidefinite program (SDP). Crucially, the size of this program is determined only by cutoff parameters, and the program depends on

N_Φ

only parametrically, allowing us to obtain numerically exact (up to solver precision) lower bounds on the gap. In order to turn these numerical bounds into mathematically rigorous results, we introduce an algorithm that can produce nearly-optimal exact interior feasible points for large rational SDPs even if the affine constraints are highly redundant and ill-conditioned. We additionally provide an algorithm that establishes the rank stability of the SDP in the

N_Φ\to \infty

limit, which is needed to certify the bound in the thermodynamic limit. Beyond establishing a Laughlin gap greater than

1/100

in the thermodynamic limit and revealing a potential method for deriving spectral gap lower bounds for other pseudopotentials, our work provides a general method for turning numerical SDP bounds into rigorous mathematical proofs, which may have wide applicability. In particular, this work demonstrates that bootstrap methods can be used to rigorously resolve major open problems in quantum many-body physics.

We consider general N-particle wave functions that have the form of a product of the Laughlin state with filling factor

1/\ell

and an analytic function of the N variables. This is the most general form of a wave function that can arise through a perturbation of the Laughlin state by external potentials or impurities, while staying in the lowest Landau level and maintaining the strong correlations of the original state. We show that the perturbation can only shift or lower the 1-particle density but nowhere increase it above a maximum value. Consequences of this bound for the response of the Laughlin state to external fields are discussed.

M

lowest eigenstates of a suitably chosen Hamiltonian acting within the lowest Landau level. The resulting ``energy projection'' is a controlled approximation to the exact lowest Landau level projection which improves with increasing

M

. It allows us to study the projected trial wave functions for system sizes close to the maximal sizes that can be reached by exact diagonalization and can be straightforwardly applied in any geometry. As a first application and test case, we study a class of trial wave functions first proposed by Girvin and Jach [Girvin and Jach, Phys. Rev. B 29, 5617 (1984)], which are modifications of the Laughlin states involving a single real parameter. While these modified Laughlin states probably represent the same universality class exemplified by the Laughlin wave functions, we show by extensive numerical work for systems on the sphere and torus that they provide a significant improvement of the variational energy, overlap with the exact wave function and properties of the entanglement spectrum.