## Undecidability of the spectral gap

Speaker:

Affiliation:

UCL

Location:

HIggs Centre Seminar Room, JCMB

Date:

Friday, March 10, 2017

Time:

13:00 to 14:00

Abstract:

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]