Gaplessness is not generic for translation-invariant spin chains

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, Phys. Rev. Lett. 119, 220504 (2017)] 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 [S. Bravyi and D. Gosset, J. Math. Phys. 56, 061902 (2015)]. 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.