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

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.