发表机构
School of Physics, Southeast University; Hefei National Laboratory, University of Science and Technology of China（东南大学物理学院; 中国科学技术大学合肥国家实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出纤维丛方法研究固定秩密度矩阵的Uhlmann几何，揭示支撑子空间的非平凡拓扑（如陈数），并建立与量子霍尔效应的联系，为混合态相变提供几何诊断。
AI 中文摘要
对于满秩密度矩阵，态空间是可缩的，Uhlmann丛在拓扑上是平凡的，其和乐不允许量子化的不变量；尽管Uhlmann相位是真正的几何量，这种拓扑贫乏性使其无法作为混合态物质鲁棒的物理诊断工具。然而，秩低于希尔伯特空间维数的固定秩混合态无处不在：子系统的约化态、不变扇区中的态以及无消相干子空间中的态，其支撑的维数都小于希尔伯特空间，且可随参数变化。这些态的几何要丰富得多：其支撑承载着真正的曲率、非阿贝尔和乐以及陈拓扑。我们直接在秩为$k$的密度矩阵流形上表述Uhlmann理论：最小纯化使固定秩层成为主$U(k)$丛的底空间，该丛的Uhlmann联络由Sylvester方程唯一确定，并且在特征标架中该联络具有闭式形式，在$k=1$、等权重和$k=N$时分别约化为Berry联络、Wilczek--Zee联络和忠实Uhlmann联络。固定秩丛继承了Grassmannian的拓扑，而不可分解的高阶陈拓扑要求全局特征线分裂的失效，在当前谱设定中这需要简并，最小实现是秩为2的Yang单极子，其量子化的第二陈数将该几何与四维量子霍尔响应联系起来。从真正的非阿贝尔和乐到无消相干Lindblad轨道，可解模型说明了以下内容：支撑子空间承载拓扑，谱权重塑造输运，而Uhlmann相位提供了这些混合态相变的直接几何特征。
英文摘要
For full-rank density matrices the state space is contractible, the Uhlmann bundle is topologically trivial, and the holonomy admits no quantized invariants; although the Uhlmann phase is genuinely geometric, this topological poverty has kept it from serving as a robust physical diagnostic of mixed-state matter. Mixed states of fixed rank below the Hilbert-space dimension, however, are ubiquitous: reduced states of constrained subsystems, states confined to invariant sectors, and states supported on decoherence-free subspaces can have a support smaller than the Hilbert space, a support that can vary with parameters. Their geometry is far richer: the support carries genuine curvature, non-Abelian holonomy, and Chern topology. We formulate Uhlmann's theory directly on the manifold of rank-$k$ density matrices: minimal purifications make the fixed-rank stratum the base of a principal $U(k)$-bundle whose Uhlmann connection is uniquely determined by a Sylvester equation, and in an eigenframe this connection takes a closed form reducing to the Berry, Wilczek--Zee, and faithful Uhlmann connections at $k=1$, at equal weights, and at $k=N$. The fixed-rank bundle inherits the topology of the Grassmannian, and non-factorizable higher Chern topology requires failure of the global eigenline splitting, which in the present spectral setting requires degeneracy, minimally realized by a rank-2 Yang monopole whose quantized second Chern number links the geometry to the four-dimensional quantum Hall response. Solvable models, from a genuinely non-Abelian holonomy to a dissipatively driven orbit whose Uhlmann holonomy is steered by the reservoir through an elliptic-integral spectral dressing, illustrate the content: the supporting subspace carries the topology, the spectral weights shape the transport, and Uhlmann holonomy provides a direct geometric probe of the resulting mixed-state structures.
Comments22 pages, 4 figures
