Certifying bulk spectral gaps with semidefinite programming

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. In this talk, I will introduce a complete family of certified upper bounds on the bulk spectral gap, obtained by solving a sequence of semidefinite programs. These bounds become arbitrarily tight as more computational resources are used, showing that the bulk spectral gap is semi-decidable. I will explain why this does not contradict undecidability results for other notions of spectral gap defined through sequences of finite systems with prescribed boundary conditions. As an application, we consider 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.