Gaplessness is not generic for translation-invariant spin chains. (arXiv:1903.00108v2 [quant-ph] UPDATED)

The existence of a spectral gap above the ground state has far-reaching
consequences for the low-energy physics of a quantum many-body system. A recent
work of Movassagh [R. Movassagh, PRL 119 (2017), 220504] shows that a spatially
random local quantum Hamiltonian is generically gapless. Here we observe that a
gap is more common for translation-invariant quantum spin chains, more
specifically, that these are gapped with a positive probability if the
interaction is of small rank. This is in line with a previous analysis of the
spin-$1/2$ case by Bravyi and Gosset. The Hamiltonians are constructed by
selecting a single projection of sufficiently small rank at random, and then
translating it across the entire chain. By the rank assumption, the resulting
Hamiltonians are automatically frustration-free and this fact plays a key role
in our analysis.

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