Undecidability of the spectral gap

Toby S. Cubitt (), David Perez-Garcia and Michael M. Wolf
Additional contact information
Toby S. Cubitt: University College London
David Perez-Garcia: Facultad de CC Matemáticas, Universidad Complutense de Madrid
Michael M. Wolf: Technische Universität München

Nature, 2015, vol. 528, issue 7581, 207-211

Abstract: 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.

Date: 2015
References: Add references at CitEc
Citations: View citations in EconPapers (5)

Downloads: (external link)
https://www.nature.com/articles/nature16059 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:nat:nature:v:528:y:2015:i:7581:d:10.1038_nature16059

Ordering information: This journal article can be ordered from
https://www.nature.com/

DOI: 10.1038/nature16059

Access Statistics for this article

Nature is currently edited by Magdalena Skipper

More articles in Nature from Nature
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().