The bulk spectral gap is semi-decidable: a convergent family of certified upper bounds

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!

Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph)

arXiv:2606.03836 [quant-ph]

(or arXiv:2606.03836v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2606.03836 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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