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Undecidability of the Spectral Gap

Toby S. Cubitt, David Pérez-García, Michael M. Wolf

(2018) 1502.04573v3 Pfeil

Abstract: We show that the spectral gap problem is undecidable. Specifically, we construct families of translationally-invariant, nearest-neighbour Hamilto-nians on a 2D square lattice of d-level quantum systems (d constant), for which determining whether the system is gapped or gapless is an undecid-able 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.