title: Undecidability of the spectral gap.
abstract: 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.
authors.affiliation: Department of Computer Science, University College London, Gower Street, London WC1E 6BT, UK.|DAMTP, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK.
authors.forename: Toby S
authors.initials: TS
authors.lastname: Cubitt
authors.affiliation: Departamento de Análisis Matemático and IMI, Facultad de CC Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid, Spain.|ICMAT, C/Nicolás Cabrera, Campus de Cantoblanco, 28049 Madrid, Spain.
authors.forename: David
authors.initials: D
authors.lastname: Perez-Garcia
authors.affiliation: Department of Mathematics, Technische Universität München, 85748 Garching, Germany.
authors.forename: Michael M
authors.initials: MM
authors.lastname: Wolf