Finite-size criteria for spectral gaps in $D$-dimensional quantum spin systems

Abstract

We generalize the existing finite-size criteria for spectral gaps of frustration-free spin systems to $D>2$ dimensions. We obtain a local gap threshold of $\frac{3}{n}$, independent of $D$, for nearest-neighbor interactions. The $\frac{1}{n}$ scaling persists for arbitrary finite-range interactions in ${\mathbb{Z}}^3$. The key observation is that there is more flexibility in Knabe's combinatorial approach if one employs the operator Cauchy-Schwarz inequality.

arXiv comment16 pages

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Cited by in corpus (5)

• Existence of a Spectral Gap in the Affleck-Kennedy-Lieb-Tasaki Model on the Hexagonal Lattice
• Improved local spectral gap thresholds for lattices of finite dimension
• Gaplessness is not generic for translation-invariant spin chains
• The spectral gap of a fractional quantum Hall system on a thin torus
• Sharp implications of AGSPs for degenerate ground spaces