In a paper from 2015 Toby S. Cubitt et al showed that the problem of determining the existence of a band gap in the excitation spectrum of a quantum many-body system, was undecidable. This result applied to atoms interacting via nearest neighbour interactions in a 2D lattice, and it was followed up by another publication which applied to 1D lattices as well.
These results were extended by Johannes Bausch et al in a 2021 paper, which demonstrated that there exist phase diagrams of many-body quantum systems which were also uncomputable.
Insofar, none of the papers mentioned provide a constructive proof of these results. I was wondering if an example Hamiltonian of a quantum many-body system had been identified yet, for which the existence of a spectral band gap was undecidable? My motivation for asking is that such a Hamiltonian could be constructed in experiment, and then a measurement could be made of the systems spectrum that which provably couldn't be solved by theory.