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SUMMARY:Undecidability of the Spectral Gap in One Dimension - Dr Johannes 
 Bausch\, DAMTP\, University of Cambridge
DTSTART:20190221T141500Z
DTEND:20190221T151500Z
UID:TALK112543@talks.cam.ac.uk
CONTACT:Katarzyna Macieszczak
DESCRIPTION:The spectral gap problem - determining whether the energy spec
 trum of a system has an energy gap above ground state\, or if there is a c
 ontinuous range of low-energy excitations - pervades quantum many-body phy
 sics. Recently\, this important problem was shown to be undecidable for qu
 antum systems in two (or more) spatial dimensions: it is provably impossib
 le to determine in general whether a system is gapped or gapless\, a resul
 t which has many unexpected consequences for the physics of such systems. 
 However\, there are many indications that one dimensional systems are simp
 ler than their higher-dimensional counterparts: for example\, they cannot 
 have thermal phase transitions or topological order\, and there exist high
 ly-effective numerical algorithms such as DMRG for gapped 1D systems\, exp
 loiting the fact that such systems obey an entropy area-law. Furthermore\,
  the spectral gap undecidability construction crucially relied on aperiodi
 c tilings\, which are easily seen to be impossible in 1D.\n\nSo does the s
 pectral gap problem become decidable in 1D? In this paper we prove this is
  not the case\, by constructing a family of 1D spin chains with translatio
 nally-invariant nearest neighbour interactions with undecidable spectral g
 ap. This not only proves that the spectral gap of 1D systems is just as in
 tractable\, but also predicts the existence of qualitatively new types of 
 complex physics in 1D spin chains. In particular\, it implies there are 1D
  systems with constant spectral gap and unique classical ground state for 
 all systems sizes up to an uncomputably large size\, whereupon they switch
  to a gapless behaviour with dense spectrum.
LOCATION:TCM Seminar Room\, Cavendish Laboratory
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