Haldane conjecture for integer-spin antiferromagnetic Heisenberg chains

Prove that every one-dimensional nearest-neighbor antiferromagnetic Heisenberg chain with integer on-site spin S has a strictly positive spectral gap (i.e., is gapped).

Background

The paper establishes universal inverse-square finite-size gap scaling for frustration-free, finite-range Hamiltonians on broad classes of graphs. In motivating the restriction to frustration-free models, the authors emphasize that many central spectral-gap questions outside this class remain unresolved.

As a canonical example, they point to the Haldane conjecture, which posits that integer-spin antiferromagnetic Heisenberg chains are gapped. This long-standing conjecture remains open and serves to illustrate why the paper focuses on frustration-free settings, where powerful techniques (e.g., detectability lemma methods) can be applied.

References

We recall that almost all mathematical investigations of the size of spectral gaps require frustration-freeness --- without this assumption even absolutely fundamental questions like the Haldane conjecture (which asserts that any integer-spin Heisenberg antiferromagnetic chain is gapped) are completely open.

— On the critical finite-size gap scaling for frustration-free Hamiltonians (2409.09685 - Lemm et al., 15 Sep 2024) in Section 1.2, Assumption on the Hamiltonian