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CATEGORIES:Theory of Condensed Matter
SUMMARY:Undecidability of the Spectral Gap in One Dimensio
n - Dr Johannes Bausch\, DAMTP\, University of Cam
bridge
DTSTART;TZID=Europe/London:20190221T141500
DTEND;TZID=Europe/London:20190221T151500
UID:TALK112543AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/112543
DESCRIPTION:The spectral gap problem - determining whether the
energy spectrum of a system has an energy gap abo
ve ground state\, or if there is a continuous rang
e of low-energy excitations - pervades quantum man
y-body physics. Recently\, this important problem
was shown to be undecidable for quantum systems in
two (or more) spatial dimensions: it is provably
impossible to determine in general whether a syste
m is gapped or gapless\, a result which has many u
nexpected consequences for the physics of such sys
tems. However\, there are many indications that on
e dimensional systems are simpler than their highe
r-dimensional counterparts: for example\, they can
not have thermal phase transitions or topological
order\, and there exist highly-effective numerical
algorithms such as DMRG for gapped 1D systems\, e
xploiting the fact that such systems obey an entro
py area-law. Furthermore\, the spectral gap undeci
dability construction crucially relied on aperiodi
c tilings\, which are easily seen to be impossible
in 1D.\n\nSo does the spectral gap problem become
decidable in 1D? In this paper we prove this is n
ot the case\, by constructing a family of 1D spin
chains with translationally-invariant nearest neig
hbour interactions with undecidable spectral gap.
This not only proves that the spectral gap of 1D s
ystems is just as intractable\, but also predicts
the existence of qualitatively new types of comple
x physics in 1D spin chains. In particular\, it im
plies there are 1D systems with constant spectral
gap and unique classical ground state for all syst
ems sizes up to an uncomputably large size\, where
upon they switch to a gapless behaviour with dense
spectrum.
LOCATION:TCM Seminar Room\, Cavendish Laboratory
CONTACT:Katarzyna Macieszczak
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