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Undecidability of the Spectral Gap in One Dimension

Thursday February 14, 2019
3:30 pm

 Presenter:  Johannes Bausch, University of Cambridge
 Series:  CQuIC Seminars
 Abstract:  The spectral gap problem - determining whether the energy
spectrum of a system has an energy gap above ground state, or if there
is a continuous range of low-energy excitations - pervades quantum
many-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 system is
gapped or gapless, a result which has many unexpected consequences for
the physics of such systems. However, there are many indications that
one dimensional systems are simpler than their higher-dimensional
counterparts: for example, they cannot have thermal phase transitions or
topological order, and there exist highly-effective numerical algorithms
such as DMRG for gapped 1D systems, exploiting the fact that such
systems obey an entropy area-law. Furthermore, the spectral gap
undecidability construction crucially relied on aperiodic tilings, which
are easily seen to be impossible in 1D.
So does the spectral 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 translationally-invariant nearest neighbour interactions
with undecidable spectral gap. This not only proves that the spectral
gap of 1D systems is just as intractable, 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.
 Host:  Elizabeth Crosson
 Location:  Room 190, Physics & Astronomy

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