aiqbench analysis
This research establishes a rigorous framework using semidefinite programs to generate certified upper bounds on the bulk spectral gap of quantum many-body systems, resolving a long-standing challenge in determining these properties without relying solely on estimates. By demonstrating that the problem is semi-decidable for this specific definition, the work provides mathematically guaranteed constraints rather than probabilistic approximations, marking a significant methodological advance over previous variational approaches. The study validates its methodology with new nontrivial bounds applied to the spin-1/2 kagome lattice Heisenberg antiferromagnet, offering fresh insights into complex quantum magnetic materials.
The bulk spectral gap is semi-decidable: a convergent family of certified upper bounds
arXiv:2606.03836v2 Announce Type: replace Abstract: Determining spectral gaps in the thermodynamic limit is a central challenge in quantum many-body physics. Existing rigorous methods are largely limited to special settings, while variational numerical approaches typically provide estimates rather than certified bounds. Here we introduce a complete family of certified upper bounds on the bulk spectral gap of quantum many-body systems. These upper bounds are obtained by solving a series of semidefinite programs and they become arbitrarily tight at the cost of more computational resources. This shows that the bulk spectral gap is semi-decidable, in contrast to undecidability results for alternative notions of spectral gap based on sequences of finite systems with prescribed boundary conditions. As a proof of principle, we apply our algorithm to the spin-$\frac{1}{2}$ kagome lattice Heisenberg antiferromagnet and obtain, to our knowledge, the first nontrivial certified upper bounds on its bulk spectral gap.
Abstract:Determining spectral gaps in the thermodynamic limit is a central challenge in quantum many-body physics. Existing rigorous methods are largely limited to special settings, while variational numerical approaches typically provide estimates rather than certified bounds. Here we introduce a complete family of certified upper bounds on the bulk spectral gap of quantum many-body systems. These upper bounds are obtained by solving a series of semidefinite programs and they become arbitrarily tight at the cost of more computational resources. This shows that the bulk spectral gap is semi-decidable, in contrast to undecidability results for alternative notions of spectral gap based on sequences of finite systems with prescribed boundary conditions. As a proof of principle, we apply our algorithm to the spin-$\frac{1}{2}$ kagome lattice Heisenberg antiferromagnet and obtain, to our knowledge, the first nontrivial certified upper bounds on its bulk spectral gap.
30 pages, 3 figures; Supplementary Information (22 pages) included. Comments welcome! v2: Corrected theorem-type cross-references by compiling with TeX Live 2023; minor wording revisions
Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph)
arXiv:2606.03836 [quant-ph]
(or arXiv:2606.03836v2 [quant-ph] for this version)
Full text under the publisher's open licence · 181 words
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