The spectral gap - the difference in energy between the ground state and the first excited state - is of central importance to quantum many-body physics. It determines the phase diagram at low temperature, with quantum phase transitions and critical phenomena occurring when the gap vanishes. Some of the most challenging and long-standing open problems in theoretical physics concern the spectral gap, such as the famous Haldane conjecture, or the infamous Yang-Mills gap conjecture (one of the Millennium Prize problems). These problems - and many others - are all particular cases of the general spectral gap problem: Given a quantum many-body Hamiltonian, is the system it describes gapped or gapless?

We prove that this problem is undecidable (in exactly the same sense as the Halting Problem was proven to be undecidable by Turing). This also implies that the spectral gap of certain quantum many-body Hamiltonians is not determined by the axioms of mathematics (in much the same sense as Goedel's incompleteness theorem implies that certain theorems are mathematically unprovable). Our results also extend to many other important low-temperature properties of quantum many-body systems, such correlation functions.

The proof is complex and draws on a wide variety of techniques, ranging from mathematical physics to theoretical computer science, from Hamiltonian complexity theory, quantum algorithms and quantum computing to fractal tilings. I will explain the result, sketch the techniques involved in the proof at an accessible level, and discuss the striking implications this may have both for theoretical physics, and for physics more generally (which, after all, happens in the laboratory not in Hilbert space!).

Based on the following papers:

Undecidability of the Spectral Gap

Toby Cubitt, David Perez-Garcia and Michael Wolf

Nature, 528, p207-211, (2015)

arXiv:1502.04135[quant-ph]

Undecidability of the Spectral Gap (full version, 143 pages)

Toby Cubitt, David Perez-Garcia and Michael Wolf

arXiv:1502.04573[quant-ph]