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Undecidability of the Spectral Gap (full version)
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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.
Forward citations
Cited by 3 Pith papers
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A Hierarchy of Spectral Gap Certificates for Frustration-Free Spin Systems
A hierarchy of SDPs yields lower bounds on spectral gaps of frustration-free Hamiltonians that encompass and improve upon Knabe's bound on 1D spin chains.
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Local Topological Quantum Order and Spectral Gap Stability for the AKLT Models on the Hexagonal and Lieb Lattices
AKLT models on hexagonal and Lieb lattices satisfy LTQO via polymer expansions, implying spectral gap stability under perturbations.
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Local Topological Quantum Order and Spectral Gap Stability for the AKLT Models on the Hexagonal and Lieb Lattices
Proves LTQO for AKLT models on hexagonal and Lieb lattices by modifying the 1988 polymer representation to obtain uniform exponential decay of boundary effects.
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