Undecidability of the spectral gap

February 8, 2016
3:00pm to 4:00pm

IFT Seminar Room/Red Room

Theoretical Physics, general interest
David Perez Garcia
U. Complutense & ICMAT

IFT Seminar Room/Red Room


The spectral gap—the energy diference between the ground state and first excited state—is central to quantum many-body physics. Many challenging open problems, such as the Haldane conjecture, existence of gapped topological spin liquid phases, or the Yang-Mills conjecture, concern spectral gaps. These and other problems are particular cases of the general spectral gap problem: given a quantum many-body Hamiltonian, is it gapped or gapless? In this work, we prove that the spectral gap problem is undecidable. We construct families of quantum spin systems on a 2D lattice with translationally-invariant, nearest-neighbour interactions for which this is an undecidable problem. This result extends to undecidability of other low energy properties, such as 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. It also implies that there exist models for which the presence or absence of a spectral gap is not defined by the axioms of mathematics.