Undecidability of the Spectral Gap (full version)

TL;DR

This paper proves that determining whether certain quantum many-body systems are gapped or gapless is an undecidable problem, highlighting fundamental limits in quantum spectral analysis.

Contribution

It establishes the undecidability of the spectral gap problem for translationally-invariant 2D quantum Hamiltonians, a significant theoretical breakthrough.

Findings

Spectral gap problem is undecidable for certain Hamiltonians.

Construction of Hamiltonians with undecidable spectral properties.

Promise of either a constant gap or continuous spectrum in the thermodynamic limit.

Abstract

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.