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 O (γ-1 /2) , one gets a Hamiltonian with an Ω (1 ) spectral gap.
Physical Review B
- Pub Date:
- May 2016
- Quantum Physics;
- Condensed Matter - Strongly Correlated Electrons
- 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