/"Our results imply that for any consistent, recursive axiomatisation of mathematics,
there exist specific Hamiltonians for which the presence or absence of a spectral gap is
independent of the axioms. These results have a number of important implications for
condensed matter and many-body quantum theory."/
Hmm. Another point at which computationalism might be able to predict something about
physics?
Brent
-------- Forwarded Message --------
1. arXiv:1502.04573 <http://arxiv.org/abs/1502.04573> [pdf
<http://arxiv.org/pdf/1502.04573>, other <http://arxiv.org/format/1502.04573>]
Undecidability of the Spectral Gap (full version)
Toby Cubitt <http://arxiv.org/find/quant-ph/1/au:+Cubitt_T/0/1/0/all/0/1>,
David
Perez-Garcia
<http://arxiv.org/find/quant-ph/1/au:+Perez_Garcia_D/0/1/0/all/0/1>,
Michael M. Wolf <http://arxiv.org/find/quant-ph/1/au:+Wolf_M/0/1/0/all/0/1>
Comments: 146 pages, 56 theorems etc., 15 figures. Read "Extended Overview"
section
for an overview, and "Structure of the paper" section for a reading guide!
See shorter
companion paper arXiv:1502.04135 <http://arxiv.org/abs/1502.04135> (same
title and
authors) for a short version omitting technical details. v2: Small but
important fix
to wording of abstract
Subjects: Quantum Physics (quant-ph); Other Condensed Matter
(cond-mat.other); High
Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
2. arXiv:1502.04135 <http://arxiv.org/abs/1502.04135> [pdf
<http://arxiv.org/pdf/1502.04135>, other <http://arxiv.org/format/1502.04135>]
Undecidability of the Spectral Gap (short version)
Toby Cubitt <http://arxiv.org/find/quant-ph/1/au:+Cubitt_T/0/1/0/all/0/1>,
David
Perez-Garcia
<http://arxiv.org/find/quant-ph/1/au:+Perez_Garcia_D/0/1/0/all/0/1>,
Michael M. Wolf <http://arxiv.org/find/quant-ph/1/au:+Wolf_M/0/1/0/all/0/1>
Comments: 8 pages, 3 figures. See long companion paper arXiv:1502.04573
<http://arxiv.org/abs/1502.04573> (same title and authors) for full
technical details
Subjects: Quantum Physics (quant-ph); Other Condensed Matter
(cond-mat.other); High
Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
http://arxiv.org/abs/1502.04573
Undecidability of the Spectral Gap (full version)
Toby Cubitt <http://arxiv.org/find/quant-ph/1/au:+Cubitt_T/0/1/0/all/0/1>, David
Perez-Garcia <http://arxiv.org/find/quant-ph/1/au:+Perez_Garcia_D/0/1/0/all/0/1>, Michael
M. Wolf <http://arxiv.org/find/quant-ph/1/au:+Wolf_M/0/1/0/all/0/1>
(Submitted on 16 Feb 2015 (v1 <http://arxiv.org/abs/1502.04573v1>), last revised 19 Feb
2015 (this version, v2))
We show that the spectral gap problem is undecidable. Specifically, we
construct
families of translationally-invariant, nearest-neighbour Hamiltonians on a
2D square
lattice of d-level quantum systems (d constant), for which determining
whether the
system is gapped or gapless is an undecidable problem. This is true even
with the
promise that each Hamiltonian is either gapped or gapless in the strongest
sense: it
is promised to either have continuous spectrum above the ground state in the
thermodynamic limit, or its spectral gap is lower-bounded by a constant in
the
thermodynamic limit. Moreover, this constant can be taken equal to the local
interaction strength of the Hamiltonian.
This implies that it is logically impossible to say in general whether a
quantum
many-body model is gapped or gapless. Our results imply that for any
consistent,
recursive axiomatisation of mathematics, there exist specific Hamiltonians
for which
the presence or absence of a spectral gap is independent of the axioms.
These results have a number of important implications for condensed matter
and
many-body quantum theory.
--
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to [email protected].
To post to this group, send email to [email protected].
Visit this group at http://groups.google.com/group/everything-list.
For more options, visit https://groups.google.com/d/optout.
