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arXiv:2610.18854

A Hierarchy of Exact Lower Bounds on the Laughlin Spectral Gap

Michael G. Scheer, Eslam Khalaf

Published Oct 13, 2026• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2610.18854

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We develop a bootstrap approach to deriving lower bounds on the spectral gap of the first Haldane pseudopotential Hamiltonian on the sphere with NΦN_ΦNΦ​ flux quanta for arbitrary NΦN_ΦNΦ​, including in the thermodynamic limit NΦ→∞N_Φ\to \inftyNΦ​→∞. 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 HHH is a Hamiltonian with ground state energy zero then the spectral gap is at least ε>0ε> 0ε>0 if and only if H2⪰εHH^2 \succeq εHH2⪰εH. To show this inequality, we split H2H^2H2 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 NΦN_ΦNΦ​ 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 NΦ→∞N_Φ\to \inftyNΦ​→∞ limit, which is needed to certify the bound in the thermodynamic limit. Beyond establishing a Laughlin gap greater than 1/1001/1001/100 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.
A recent preprint by OpenAI resolved the spherical fermionic Laughlin spectral-gap conjecture for the full V1V_1V1​ interaction by a Fock-space inequality: HQ2≥γHQH_Q^2\ge γH_QHQ2​≥γHQ​ on the lowest-Landau-level Fock space for all large flux QQQ and every γγγ below 0.0461673…0.0461673\ldots0.0461673…, so that the gap above the Laughlin state at filling 1/31/31/3 is at least 1/251/251/25. 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 0.06010.06010.0601 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 HQ2≥γHQH_Q^2\geγH_QHQ2​≥γHQ​ for every γ<γ∗=0.2190037845063196…γ<γ_*=0.2190037845063196\ldotsγ<γ∗​=0.2190037845063196…, certified by exact rational arithmetic. Consequently the gap above the Laughlin state is at least 1/51/51/5 for all sufficiently large spheres, and the untruncated planar V1V_1V1​ Hamiltonian has positive spectrum bounded below by γ∗γ_*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 0.2220.2220.222, while exact diagonalization of finite systems places the true gap near 0.40.40.4. 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.
Determining spectral gaps in the thermodynamic limit is a central challenge in quantum many-body physics. When a fixed boundary condition is imposed, this problem is undecidable: no algorithm, even with unlimited resources, can decide for every Hamiltonian whether it is gapped. Yet the thermodynamic limit is meant to capture the intrinsic properties of very large systems, which should not depend on how their boundaries are chosen. We therefore consider the bulk spectral gap, which concerns only the intrinsic excitations of the infinite system, and ask what can and cannot be computed about it. First, we give a general algorithm producing a hierarchy of certified upper bounds on the bulk gap, arbitrarily tight at the cost of more computation. Second, we prove that the spectral gap problem remains undecidable in the bulk, already for translationally invariant nearest-neighbour chains of fixed local dimension. Hence no general algorithm produces convergent lower bounds, and no convergent upper-bounding algorithm, ours included, admits a computable convergence rate. The bulk spectral gap is thus certifiable from above but uncomputable from below; more precisely, the bulk-gaplessness promise problem is RE-complete, exactly as hard as the halting problem. The same construction shows that no general algorithm can decide whether a given product state is a ground state, and that convergent bounds on ground-state expectation values admit no computable convergence rate in general. Finally, we obtain the first nontrivial certified upper bounds on the bulk gap of the spin-1/2 kagome Heisenberg antiferromagnet.
Estimating spectral gaps of quantum many-body Hamiltonians is a highly challenging computational task, even under assumptions of locality and translation-invariance. Yet, the quest for rigorous gap certificates is motivated by their broad applicability, ranging from many-body physics to quantum computing and classical sampling techniques. Here we present a general method for obtaining lower bounds on the spectral gap of frustration-free quantum Hamiltonians in the thermodynamic limit. We formulate the gap certification problem as a hierarchy of optimization problems (semidefinite programs) in which the certificate – a proof of a lower bound on the gap – is improved with increasing levels. Our approach encompasses existing finite-size methods, such as Knabe's bound and its subsequent improvements, as those appear as particular possible solutions in our optimization, which is thus guaranteed to either match or surpass them. We demonstrate the power of the method on one-dimensional spin-chain models where we observe an improvement by several orders of magnitude over existing finite size criteria in both the accuracy of the lower bound on the gap, as well as the range of parameters in which a gap is detected.
Determining correlated ground states directly in the thermodynamic limit remains a central challenge in quantum many-body physics. Here we develop a quantum many-body bootstrap that yields a rigorous and tight lower bound on the thermodynamic ground-state energy of a fractional quantum Hall system together with an optimized reduced density matrix ρW⋆ρ_W^\starρW⋆​ on a spatial window. For sufficiently tight bounds, we find that ρW⋆ρ_W^\starρW⋆​ accurately reproduces the ground-state reduced density matrix, as directly tested by its fidelity with known model states. For exactly solvable Laughlin and Moore--Read parent Hamiltonians, the bootstrap also reproduces pair correlations and orbital entanglement spectra (OES). For Landau-level-projected Coulomb interactions across multiple fillings, the energy bounds closely track established thermodynamic estimates, while the window density matrix recovers the familiar ν=1/3ν=1/3ν=1/3 Laughlin and ν=5/2ν=5/2ν=5/2 Moore--Read states and the composite Fermi liquid at ν=1/2ν=1/2ν=1/2. More importantly, the same framework also resolves unsettled regimes. At ν=1/4ν=1/4ν=1/4, the OES and real-space correlations identify a composite Fermi liquid, while at ν=12/5ν=12/5ν=12/5 the combined diagnostics suggest RR‾3\overline{\mathrm{RR}}_3RR3​ as the ground-state order. Our bootstrap therefore provides direct access to both ground-state energies and many-body structure in the thermodynamic limit.
Proving spectral gaps above the ground space is a central problem in quantum many-body physics. Yet, finding lower bounds on the gap is notoriously difficult. We revisit the martingale method, originally developed by Fannes, Nachtergaele, and Werner [Fannes '92, Nachtergaele '96] to prove the existence of spectral gaps of parent Hamiltonians of Matrix Product States (MPS), and provide improvements to the different steps of the method. Most importantly, we devise a new technique which allows to compute the key quantity in the martingale method -- the overlap of local ground spaces -- exactly and efficiently. This enables a clear improvement of the method, allowing it to outperform other existing techniques to lower bound gaps, which we demonstrate by benchmarking on several models. Remarkably, our -- numerically motivated -- approach at the same time also yields a significantly simplified proof of the fact that parent Hamiltonians of any (well-behaved) MPS are always gapped.
The bootstrap method aims to solve problems by imposing constraints on the space of physical observables, which often follow from physical assumptions such as positivity and symmetry. Here, we employ a bootstrap approach to study interacting electrons in the lowest Landau level by minimizing the energy as a function of the static structure factor subject to a set of constraints, bypassing the need to construct the full many-body wave function. This approach rigorously lower bounds the ground state energy, making it complementary to conventional variational upper bounds. We show that the lower bound we obtain is relatively tight, within at most a few percent from the ground state energy computed with exact diagonalization at small system sizes, and generally gets tighter as we include more constraints. We also show that, by combining the bootstrap lower bounds with variational Monte Carlo calculations for various quantum Hall states, we can obtain two-sided bounds on the ground state energy that rigorously bounds the error to below a few percentage for large system sizes where exact diagonalization is not accessible. Beyond the energetics, our results reproduce the correct power law dependence of the pair correlation function at short distances and the existence of a large entanglement gap in the two-particle entanglement spectra for the Laughlin states at ν = 1 / 3 . We further identify signatures of the composite Fermi liquid state close to half filling. This shows that the bootstrap approach is capable, in principle, of describing nontrivial gapped topologically ordered, as well as gapless, phases. We further study the evolution of the ground state energy and correlation functions as we interpolate between the lowest and first Landau levels at half filling and find indications of a phase transition from a composite Fermi liquid to a non-Abelian state. We also discuss more generally how the geometry of the allowed set of structure factors can be used to diagnose phase transitions. At the end, we discuss possible extensions and current limitations of this approach and how they can be overcome in future studies. Our work establishes numerical bootstrap as a promising method to study many-body phases in topological bands, paving the way to its application in moiré platforms where the energetic competition between fractional quantum anomalous Hall, symmetry broken, and gapless states remains poorly understood.