Undecidability of the Spectral Gap (short version)
Abstract
The spectral gap  the energy difference between the ground state and first excited state  is central to quantum manybody physics. Many challenging open problems, such as the Haldane conjecture, existence of gapped topological spin liquid phases, or the YangMills gap conjecture, concern spectral gaps. These and other problems are particular cases of the general spectral gap problem: given a quantum manybody Hamiltonian, is it gapped or gapless? Here we prove that this is an undecidable problem. We construct families of quantum spin systems on a 2D lattice with translationallyinvariant, nearestneighbour interactions for which the spectral gap problem is undecidable. This result extends to undecidability of other low energy properties, such as existence of algebraically decaying groundstate correlations. The proof combines Hamiltonian complexity techniques with aperiodic tilings, to construct a Hamiltonian whose ground state encodes the evolution of a quantum phaseestimation 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. It also implies that there exist models for which the presence or absence of a spectral gap is independent of the axioms of mathematics.
 Publication:

arXiv eprints
 Pub Date:
 February 2015
 arXiv:
 arXiv:1502.04135
 Bibcode:
 2015arXiv150204135C
 Keywords:

 Quantum Physics;
 Condensed Matter  Other Condensed Matter;
 High Energy Physics  Theory;
 Mathematical Physics
 EPrint:
 v2: published version. 7 pages, 3 figures. See long companion paper arXiv:1502.04573 (same title and authors) for full technical details. v3: Added supplementary material as ancilliary file (previously part of arXiv:1502.04573v2)