RT Journal Article
T1 Undecidability of the spectral gap
A1 Cubbit, Toby S.
A1 Pérez García, David
A1 Wolf, Michael M.
AB 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.
PB Nature Publishing Group
SN 0028-0836
YR 2015
FD 2015-12-10
LK https://hdl.handle.net/20.500.14352/24283
UL https://hdl.handle.net/20.500.14352/24283
LA eng
NO Supplementary material: http://eprints.sim.ucm.es/38062/
NO Comunidad de Madrid
NO Royal Society
NO Ministerio de Economía y Competitividad (MINECO)
NO European Research Council (ERC) under the European Union
NO John Templeton Foundation
DS Docta Complutense
RD 4 abr 2025