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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.

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    $\begingroup$ About your motivation: I quote the last sentence of the abstract of the Bausch 2020 paper you cite, "In particular, it implies there are 1D systems with a constant spectral gap and nondegenerate classical ground state for all systems sizes up to an uncomputably large size, whereupon they switch to a gapless behavior with dense spectrum." Presumably, "uncomputably" here could just as well be replaced by "unconstructably", so it's not so clear that the experiment would exist that would provide the paradox you're implying. $\endgroup$
    – Michael Engelhardt
    Commented Oct 4, 2021 at 4:37

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