Undecidability of the Spectral Gap (full version)

Toby Cubitt, David Perez-Garcia, Michael M. Wolf

Published 2015-02-16Version 1

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. This implies that it is logically impossible to say in general whether a quantum many-body system is gapped or gapless. Our results imply that there exist specific Hamiltonians for which the presence or absence of a spectral gap is independent of all consistent axiomatisations of mathematics. These results have a number of important implications for condensed matter and many-body quantum theory.