格拉斯曼流形上的量子态投影算子：几何、和乐与拓扑
Quantum-State Projectors on Grassmannian: Geometry, Holonomy, and Topology
AI总结：
该研究针对量子系统的带隙哈密顿量，以全局投影算子的微分dP为核心，构建格拉斯曼流形上的最短路径，推导多带威尔逊圈行列式的基无关表达式，统一局域量子几何、和乐与拓扑，避免全局规范固定。
AI中文摘要：
N能级量子系统中孤立的k个能带构成秩为k的谱投影算子，由此得到复格拉斯曼流形Gr(k,N)中的一个映射。对于光滑带隙哈密顿量，即便能带拓扑阻碍了全局光滑周期或对称兼容的布洛赫标架，该投影算子仍为全局光滑且周期的。我们将全局定义的微分dP作为核心对象：它是格拉斯曼流形映射的切场，消除了所选子空间内的非物理旋转，保留了物理的带间跃迁。该切数据的带间块同时决定量子度量与贝里曲率；相关水平生成元产生有限格拉斯曼流形运动，而dP的楔积则纳入拓扑形式。我们构造了环境格拉斯曼流形中两个投影算子间的最短路径，并证明其水平生成元块的奇异值是端点子空间间的主夹角。该构造为离散几何相位提供了分段测地解释。在另一项研究中，我们从有序投影算子乘积的幂次迹中推导了多带威尔逊圈行列式的与基无关表达式，无需将简并能带多重态分解为单个能带。最后，相同的切向量演算整理了陈示性类、手征缠绕数和时间反演Z2指标。所得框架统一了局域量子几何、有限子空间距离、和乐与拓扑，同时避免了全局规范固定。
英文摘要：
An isolated group of $k$ bands in an $N$-level quantum system defines a rank-$k$ spectral projector and hence a map into the complex Grassmannian $\mathrm{Gr}(k,N)$. For a smooth gapped Hamiltonian, this projector is globally smooth and periodic even when band topology obstructs a globally smooth periodic, or symmetry-compatible, Bloch frame. We take the globally defined differential $\dd P$ as the central object: it is the tangent field of the Grassmannian map, removes unphysical rotations within the selected subspace, and retains the physical interband transition. The interband block of this tangent data simultaneously determines the quantum metric and Berry curvature; the associated horizontal generator produces finite Grassmannian motion, while wedge products of $\dd P$ enter topological forms. We construct a shortest path between two projectors in the ambient Grassmannian and show that the singular values of its horizontal generator block are the principal angles between the endpoint subspaces. This construction provides a piecewise-geodesic interpretation of discrete geometric phases. In a separate development, we derive a basis-independent expression for the determinant of a multiband Wilson loop from traces of powers of an ordered projector product, without decomposing a degenerate band multiplet into individual bands. Finally, the same tangent-vector calculus organizes Chern characters, chiral winding numbers, and the time-reversal $\mathbb Z_2$ index. The resulting framework unifies local quantum geometry, finite subspace distance, holonomy, and topology while avoiding global gauge fixing.
