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The aim is to investigate a certification method for spectral gaps in frustration-free spin chains and relate it to graph acyclicity.

August 6, 2026Open Access

A fixed-level certification dichotomy for an infinite family of frustration-free spin chains

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The aim is to investigate a certification method for spectral gaps in frustration-free spin chains and relate it to graph acyclicity.
Defined a relationship between directed graphs and spin-chain interactions by assigning penalties and hopping bands.
Proved a three-term equivalence involving acyclic graphs, spectral gaps, and certification values using an explicit rational certificate.
Conducted a free-fermion analysis to quantify cyclic behavior in the spin chains.
Demonstrated that acyclicity condition ensures a positive spectral gap.
Provided a specific bound for the local translation-invariant certification at level 5 of min{1/48, 1/sqrt(12H)}.
Computed the exact finite-volume gaps for directed cycles, yielding specific mathematical expressions.

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This preprint establishes a fixed-level certification for spectral gaps in an infinite family of spin chains, highlighting acyclicity's role.

Guillaume Lecomte (2026) studied this question.

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This preprint establishes a fixed-level certification for spectral gaps in an infinite family of spin chains, highlighting acyclicity's role.