标题： 密度矩阵几何和求和规则 显示英文标题

标题： Density Matrix Geometry and Sum Rules

摘要： 几何在从异常输运系数到其相关求和规则的广泛物理响应中起着基础作用。 显著的例子包括霍尔电导率的量子化和Souza-Wilkens-Martin(SWM)求和规则——两者在零温度下有效，与相互作用和无序无关。 SWM求和规则的有限温度推广已在文献中被探讨，揭示了与密度矩阵几何的深刻联系。 基于时间依赖几何框架的最新进展，我们提出了用于热密度矩阵的时间依赖量子几何张量。 这一形式提供了在涨落-耗散定理框架内对已知求和规则的统一解释，进一步阐明了它们的基本几何起源。 此外，它还提供了在零温度以外的实验可探测方法来研究量子几何。 显示英文摘要

摘要： Geometry plays a fundamental role in a wide range of physical responses, from anomalous transport coefficients to their related sum rules. Notable examples include the quantization of the Hall conductivity and the Souza-Wilkens-Martin (SWM) sum rule -- both valid at zero temperature, independent of interactions and disorder. The finite-temperature generalization of the SWM sum rule has been explored in the literature, revealing deep connections to the geometry of density matrices. Building on recent advances in time-dependent geometric frameworks, we propose a time-dependent quantum geometric tensor for thermal density matrices. This formalism provides a unified interpretation of known sum rules within the framework of the fluctuation-dissipation theorem, further elucidating their fundamental geometric origin. In addition, it provides experimentally accessible methods to probe quantum geometry beyond the zero-temperature regime.