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Undecidability of the spectral gap


The spectral gap—the energy difference between the ground state and first excited state of a system—is central to quantum many-body physics. Many challenging open problems, such as the Haldane conjecture, the question of the existence of gapped topological spin liquid phases, and the Yang–Mills gap conjecture, concern spectral gaps. These and other problems are particular cases of the general spectral gap problem: given the Hamiltonian of a quantum many-body system, is it gapped or gapless? Here we prove that this is an undecidable problem. Specifically, we construct families of quantum spin systems on a two-dimensional lattice with translationally invariant, nearest-neighbour interactions, for which the spectral gap problem is undecidable. This result extends to undecidability of other low-energy properties, such as the existence of algebraically decaying ground-state correlations. The proof combines Hamiltonian complexity techniques with aperiodic tilings, to construct a Hamiltonian whose ground state encodes the evolution of a quantum phase-estimation algorithm followed by a universal Turing machine. The spectral gap depends on the outcome of the corresponding ‘halting problem’. Our result implies that there exists no algorithm to determine whether an arbitrary model is gapped or gapless, and that there exist models for which the presence or absence of a spectral gap is independent of the axioms of mathematics.

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Figure 1: Gapped and gapless systems.
Figure 2: Relating ground state energy density to spectral gap.
Figure 3: Complete Hamiltonian construction.

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T.S.C. thanks IBM. T. J. Watson Laboratory for their hospitality, and C. Bennett in particular for discussions about this work. T.S.C., D.P.-G. and M.M.W. thank the Isaac Newton Institute for Mathematical Sciences, Cambridge for their hospitality during the programme “Mathematical Challenges in Quantum Information”, where part of this work was carried out. T.S.C. is supported by the Royal Society. D.P.G. acknowledges support from MINECO (grant MTM2011-26912 and PRI-PIMCHI-2011-1071), Comunidad de Madrid (grant QUITEMAD+-CM, ref. S2013/ICE-2801) and the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 648913). This work was made possible through the support of grant no. 48322 from the John Templeton Foundation. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation.

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Correspondence to Toby S. Cubitt.

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Cubitt, T., Perez-Garcia, D. & Wolf, M. Undecidability of the spectral gap. Nature 528, 207–211 (2015).

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