标题： 厄米和非厄米拓扑转变的流形距离表征 显示英文标题

标题： Hermitian and Non-Hermitian Topological Transitions Characterized by Manifold Distance

摘要： 拓扑相通常由整数表示的拓扑不变量来表征。 然而，不同的拓扑系统通常需要不同的拓扑不变量来测量，这些定义在临界点通常失效。 因此，预测两种不同拓扑相之间的转换会发生什么具有挑战性。 为了解决这些问题，我们提出了一种基于量子信息理论中的保真度和迹距离的一般定义：流形距离（MD）。 这种定义不依赖于贝里连接，而是依赖于两个流形的信息——它们的基态波函数。 因此，它可以测量不同的拓扑系统（包括传统的能带拓扑模型、非厄米系统和无能隙系统等），并在两种拓扑相之间的转换中表现出一些普遍规律。 我们的研究显示，对于不同的拓扑流形，MD的变化率（一阶导数）或易感性（二阶导数）在临界点附近表现出各种发散行为。 与可用于诊断一维和二维中短程纠缠态的奇异关联器相比，MD更加通用，可以应用于非厄米系统和长程纠缠态。 对于后续研究，我们期望该方法能够推广到实空间或非格点模型，以便促进对更广泛的物理平台（如开放系统和多体局域化）的研究。 显示英文摘要

摘要： Topological phases are generally characterized by topological invariants denoted by integer numbers. However, different topological systems often require different topological invariants to measure, and theses definition usually fail at critical points. Therefore, it's challenging to predict what would occur during the transformation between two different topological phases. To address these issues, we propose a general definition based on fidelity and trace distance from quantum information theory: manifold distance (MD). This definition does not rely on the berry connection but rather on the information of the two manifolds - their ground state wave functions. Thus, it can measure different topological systems (including traditional band topology models, non-Hermitian systems, and gapless systems, etc.) and exhibit some universal laws during the transformation between two topological phases. Our research demonstrates for different topological manifolds, the change rate (first-order derivative) or susceptibility (second-order derivative) of MD exhibit various divergent behaviors near the critical points. Compared to the strange correlator, which could be used as a diagnosis for short-range entangled states in 1D and 2D, MD is more universal and could be applied to non-Hermitian systems and long-range entangled states. For subsequent studies, we expect the method to be generalized to real-space or non-lattice models, in order to facilitate the study of a wider range of physical platforms such as open systems and many-body localization.