A fascinating application of computability theory to physics:
Undecidability of the Spectral Gap
Toby Cubitt, David Perez-Garcia, and Michael M. Wolf
The spectral gap—the difference in energy between the ground state and
the first excited state—is one of the most important prop-
erties of a quantum many-body system. Quantum phase transitions occur
when the spectral gap vanishes and the system becomes
critical. Much of physicsis concerned with understanding the phase
diagrams of quantum systems, and some of the most challenging
and long-standing open problems in theoretical physics concern the
spectral gap, 1–3 such as the Haldane conjecture 4 that the Heisen-
berg chain is gapped for integer spin, proving existence of a gapped
topological spin liquid phase, 2,3 or the Yang-Mills gap conjecture 5
(one of the Millennium Prize problems). These problems are all
particular cases of the general spectral gap problem: Given a quan-
tum many-body Hamiltonian, is the system it describes gapped or gapless?
Here we show that this problem is undecidable, in the
same sense as the Halting Problem was proven to be undecidable by
Turing. 6 A consequence of this is that the spectral gap of certain
quantum many-body Hamiltonians is not determined by the axioms of
mathematics, much as Gödels incompleteness theorem implies
that certain theorems are mathematically unprovable. We extend these
results to prove undecidability of other low temperature prop-
erties, such as correlation functions. The proof hinges on simple
quantum many-body models that exhibit highly unusual physics in
the thermodynamic limit.
arXiv:1502.04135v1 [quant-ph] 13 Feb 2015
Brent
--
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 everything-list+unsubscr...@googlegroups.com.
To post to this group, send email to everything-list@googlegroups.com.
Visit this group at http://groups.google.com/group/everything-list.
For more options, visit https://groups.google.com/d/optout.
