We show that the spectral gap problem is undecidable. Specifically, we
construct families of translationally-invariant, nearest-neighbour Hamiltonians
on a 2D square lattice of d-level quantum systems (d constant), for which
determining whether the system is gapped or gapless is an undecidable problem.
This is true even with the promise that each Hamiltonian is either gapped or
gapless in the strongest sense: it is promised to either have continuous
spectrum above the ground state in the thermodynamic limit, or its spectral gap
is lower-bounded by a constant in the thermodynamic limit. Moreover, this
constant can be taken equal to the local interaction strength of the
Hamiltonian.Comment: v1: 146 pages, 56 theorems etc., 15 figures. See shorter companion
paper arXiv:1502.04135 (same title and authors) for a short version omitting
technical details. v2: Small but important fix to wording of abstract. v3:
Simplified and shortened some parts of the proof; minor fixes to other parts.
Now only 127 pages, 55 theorems etc., 10 figures. v4: Minor updates to
introductio