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.
From: Xiangling Xu [view email]
[v1]
Tue, 2 Jun 2026 16:18:19 UTC (563 KB)
[v2]
Fri, 31 Jul 2026 07:24:53 UTC (563 KB)
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