Gaplessness is not generic for translation-invariant spin chains

Abstract

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.

arXiv comment17 pages; minor changes; comments welcome

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• Undecidability of the Spectral Gap in One Dimension
• Correlation length in random MPS and PEPS
• Improved local spectral gap thresholds for lattices of finite dimension
• Typical Correlation Length of Sequentially Generated Tensor Network States
• Quantitatively improved finite-size criteria for spectral gaps
• Random translation-invariant Hamiltonians and their spectral gaps
• Area law for steady states of detailed-balance local Lindbladians
• A Hierarchy of Spectral Gap Certificates for Frustration-Free Spin Systems
• Spectral statistics and energy-gap scaling in $k-$local spin Hamiltonians