拓扑指标在$\mathrm{U}(1)\rtimes\mathbb{Z}_2$对称量子自旋链中的等价性
The equivalence of topological indices for $\mathrm{U}(1)\rtimes\mathbb{Z}_2$-symmetric quantum spin chains
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中文总结 AI 辅助
该研究证明了在$\mathrm{U}(1)\rtimes\mathbb{Z}_2$对称量子自旋链中Ogata指标与Tasaki扭转指标的等价性，并给出了可计算的公式，应用于反铁磁海森堡链得到Ogata指标为$(-1)^S$。
中文摘要 AI 辅助
量子自旋链中的对称保护拓扑（SPT）相通过拓扑指标来区分。Ogata的算子代数指标适用于广泛的对称类，无需连续对称性，而Tasaki的基本扭转指标则需要$\mathrm{U}(1)$对称性。我们在具有格点$\mathrm{U}(1)\rtimes\mathbb{Z}_2$对称性和对称局部唯一带隙基态的整数自旋链这一共同设定下证明了它们的等价性，且不假设平移不变性或矩阵乘积表示。电荷涨落估计给出了半链旋转的强极限构造，并固定了其相位。一个精确的群交换子恒等式将Ogata指标与局部扭转期望的极限联系起来，并给出了显式误差界。当间隙下界和相关的基态期望可用时，这提供了一个可计算的公式。作为应用，先前的扭转指标计算在显式唯一性和边界场间隙假设下，确定了反铁磁海森堡链的Ogata指标为$(-1)^S$。作者提出了问题并给出了基本策略，而ChatGPT对证明的发展做出了实质性贡献。
英文摘要
Symmetry-protected topological (SPT) phases in quantum spin chains are distinguished by topological indices. Ogata's operator-algebraic indices apply to a broad range of symmetry classes without requiring continuous symmetry, whereas the elementary twist index of Tasaki requires $\mathrm{U}(1)$ symmetry. We prove their equivalence in the common setting of integer-spin chains with on-site $\mathrm{U}(1)\rtimes\mathbb{Z}_2$ symmetry and a symmetric locally-unique gapped ground state, without assuming translation invariance or a matrix product representation. A charge-fluctuation estimate yields a strong-limit construction of half-chain rotations with their phases fixed. An exact group commutator identity then identifies the Ogata index with the limit of local twist expectations and gives an explicit error bound. This provides a computable formula when a lower bound on the gap and the relevant ground-state expectations are available. As an application, a previous twist-index calculation determines the Ogata index to be $(-1)^S$ for the antiferromagnetic Heisenberg chain under explicit uniqueness and boundary-field gap assumptions. The author formulated the problem and proposed the basic strategy, while ChatGPT made substantial contributions to the development of the proof.
发表机构
- Department of Physics, Gakushuin University（学习院大学物理系）
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