Undecidability of the spectral gap

Toby Cubitt
HIggs Centre Seminar Room, JCMB
Friday, March 10, 2017 - 13:00 to 14:00


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)
Undecidability of the Spectral Gap (full version, 143 pages)
Toby Cubitt, David Perez-Garcia and Michael Wolf