发表机构
Seoul National University（首尔大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究将理想量子几何从陈带的阿贝尔几何推广到时间反演对称量子物质的四元数非阿贝尔几何，揭示了克拉默斯对能带的四元数几何特性及相关量子结构。
AI 中文摘要
量子几何揭示了布洛赫波函数的形状如何调控关联量子现象，其标准形式描述孤立复能带，其中贝里曲率为阿贝尔型，理想几何为凯勒几何。然而，具有自旋的时间反演对称晶体需要不同的描述语言，因为克拉默斯简并对布洛赫态的配对会使贝里曲率变为非阿贝尔SU(2)场。本文证明克拉默斯对能带几何是四元数的：一个最小克拉默斯对定义了到四元数射影空间的映射，其四元数量子几何张量将量子度量与三个SU(2)贝里曲率分量统一起来；该张量的非负性施加了局域度量-曲率不等式，其饱和定义了理想陈带的非阿贝尔对应物；在四维中，理想极限进一步产生与四维量子霍尔效应相关的代数结构。研究结果将理想量子几何从陈带的阿贝尔几何推广到时间反演对称量子物质的四元数非阿贝尔几何。
英文摘要
Quantum geometry reveals how the shape of Bloch wave functions governs correlated quantum phenomena. Its standard formulation describes isolated complex bands, where Berry curvature is Abelian and ideal geometry is Kähler. However, time reversal symmetric crystals with spin require a different language since Kramers degeneracy pairs Bloch states and turns Berry curvature into a non-Abelian SU(2) field. Here we show that Kramers pair band geometry is quaternionic. A minimal Kramers pair defines a map into quaternion projective space, and its quaternionic quantum geometric tensor unifies the quantum metric with the three SU(2) Berry curvature components. The non-negativity of this tensor imposes local metric-curvature inequalities, whose saturation defines the non-Abelian counterpart of ideal Chern bands. In four dimensions, the ideal limit further yields an algebraic structure related to the four-dimensional quantum Hall effect. Our results promote ideal quantum geometry from the Abelian geometry of Chern bands to the quaternionic, non-Abelian geometry of time reversal symmetric quantum matter.
Comments38 pages, 4 figures
