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arXiv · 2606.03836

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

Xiangling Xu·Matthias Schötz·Jie Wang·Victor Magron·Igor Klep·Omar Fawzi·Marc-Olivier Renou
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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.

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Xiangling Xu, Matthias Schötz, Jie Wang, Victor Magron, Igor Klep, Omar Fawzi, Marc-Olivier Renou. 2026-06-02. The bulk spectral gap is semi-decidable: a convergent family of certified upper bounds. https://arxiv.org/abs/2606.03836

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