A Hierarchy of Exact Lower Bounds on the Laughlin Spectral Gap
Abstract
We develop a bootstrap approach to deriving lower bounds on the spectral gap of the first Haldane pseudopotential Hamiltonian on the sphere with NΦ flux quanta for arbitrary NΦ, including in the thermodynamic limit NΦ→ ∞. 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 H2 ⪰ εH. To show this inequality, we split H2 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 NΦ 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→ ∞ 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.
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BibTeX
@misc{scheer2026hierarchy,
title = {{A Hierarchy of Exact Lower Bounds on the Laughlin Spectral Gap}},
author = {Michael G. Scheer and Eslam Khalaf},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18854},
eprint = {2610.18854},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18854},
}
RIS
TY - UNPB
TI - A Hierarchy of Exact Lower Bounds on the Laughlin Spectral Gap
AU - Michael G. Scheer
AU - Eslam Khalaf
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.18854
UR - https://arxiv.org/abs/2610.18854
AB - 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→ ∞. 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 ⪰ 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→ ∞ 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.
ER -