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Undecidability of the Spectral Gap

Toby Cubitt ( UCL )

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!).

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