Undecidability of the Spectral Gap in One Dimension

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• Dr Johannes Bausch, DAMTP, University of Cambridge
• Thursday 21 February 2019, 14:15-15:15
• TCM Seminar Room, Cavendish Laboratory.

If you have a question about this talk, please contact Katarzyna Macieszczak.

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.

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