A simple proof of the detectability lemma and spectral gap amplification

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 new, 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 $\gamma >0$ to a length scale $O({\gamma }^{-1/2})$, one gets an Hamiltonian with an $\Omega (1)$ spectral gap.

9 pages, plain LaTex, 3 figures. Replaced the converse of the DL by a recent result by J. Gao, and added a new tightness result

References in corpus (2)

• Lieb-Robinson Bounds and the Exponential Clustering Theorem
• Correlation Length versus Gap in Frustration-Free Systems

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