Simple proof of the detectability lemma and spectral gap amplification

Anurag Anshu, Itai Arad, Thomas Vidick

Research output: Contribution to journalArticlepeer-review

Abstract

The detectability lemma is a useful tool for probing the structure of gapped ground states of frustration-free Hamiltonians of lattice spin models. The lemma provides an estimate on the error incurred by approximating the ground space projector with a product of local projectors. We provide a simpler proof for the detectability lemma which applies to an arbitrary ordering of the local projectors, and show that it is tight up to a constant factor. As an application, we show how the lemma can be combined with a strong converse by Gao to obtain local spectral gap amplification: We show that by coarse graining a local frustration-free Hamiltonian with a spectral gap γ>0 to a length scale Oγ-1/2, one gets a Hamiltonian with an Ω1 spectral gap.

Original languageEnglish
Article number205142
JournalPhysical Review B
Volume93
Issue number20
DOIs
StatePublished - 23 May 2016
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Electronic, Optical and Magnetic Materials
  • Condensed Matter Physics

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