PIRSA:18100004 Scientific Series Quantum Information

DOI10.48660/18100004

Series: Perimeter Institute Quantum Discussions

Lucia, A. (2018). Undecidability of the spectral gap in one dimension. Perimeter Institute. https://pirsa.org/18100004

Lucia, Angelo. Undecidability of the spectral gap in one dimension. Perimeter Institute, Oct. 24, 2018, https://pirsa.org/18100004 

@misc{ pirsa_PIRSA:18100004,
  doi = {10.48660/18100004},
  url = {https://pirsa.org/18100004},
  author = {Lucia, Angelo},
  keywords = {Quantum Information},
  language = {en},
  title = {Undecidability of the spectral gap in one dimension},
  publisher = {Perimeter Institute},
  year = {2018},
  month = {oct},
  note = {PIRSA:18100004 see, \url{https://pirsa.org}}
}

Abstract

The spectral gap problem consist in deciding, given a local interaction, whether the corresponding translationally invariant Hamiltonian on a lattice has a spectral gap independent of the system size or not. In the simplest case of nearest-neighbour frustration-free qubit interactions, there is a complete classification. On the other extreme, for two (or higher) dimensional models with nearest-neighbour interactions this problem can be reduced to the Halting Problem, and it is therefore undecidable.

There are a lot of indications that one dimensional spin chain are relatively simpler than their counterparts in higher dimensions. Nonetheless, I will present a construction of a family of nearest-neighbour, translationally invariant Hamiltonians on a spin chain, for which the spectral gap problem is undecidable.

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