# Undecidability of the Spectral Gap

- 14:00 4th December 2015 ( week 8, Michaelmas Term 2015 )Lecture Theatre B

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. 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 $1

million 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). The results 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 theoretical computer science result may have both for

theoretical physics, and for physics more generally (which, after all,

happens in the laboratory not in Hilbert space!).