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arXiv:2610.34330

Boundary-condition independence of infinite-chain ground states in the $S=1$ $J$-$D$ antiferromagnetic chain

Hal Tasaki

math-phcond-mat.stat-mechquant-phSubmitted Oct 25, 2026• Announced Oct 27, 2026• 0 citations
v1 · Oct 25, 2026v2 · Oct 27, 2026
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Abstract

Extending the analytic machinery and using the numerical inputs from OpenAI's proofs of the Haldane gap, we study infinite-chain ground states of the $S=1$ $J$-$D$ antiferromagnetic model $H=\sum_j\{\boldsymbol S_j\cdot\boldsymbol S_{j+1}+D(S_j^{\mathrm z})^2\}$ for sufficiently small $|D|$. We consider two finite-volume settings: periodic chains with an even number of sites, and open chains with an odd number of sites and the same positive magnetic field in the $\mathrm z$ direction at both endpoints. We prove that the ground states in each setting have an infinite-chain limit along the full sequence, without passing to a subsequence, and that the two limiting states coincide. Convergence is uniform in $D$ on a fixed neighborhood of zero. The common limit is a symmetric locally-unique gapped ground state, and its local expectation values depend continuously on $D$. These conclusions concern the two specified constructions, not uniqueness among all infinite-chain ground states. They provide the input for the study of a topological phase transition in a forthcoming revision of our companion paper. The author formulated the problem and proposed the basic strategy. ChatGPT Astra made substantial contributions to the inductive estimates and to completing the proof.

Comments
32 pages, 2 figures

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