Topological phase of matter is now a mainstream of research in condensed matter physics, of which the classification, synthesis, and detection of topological states have brought excitements over the recent decade while remain incomplete with ongoing challenges in both theory and experiment. Here we propose to establish a universal non-equilibrium characterization of the equilibrium topological quantum phases classified by integers, and further propose the high-precision dynamical schemes to detect such states. The framework of the dynamical classification theory consists of basic theorems. First, we uncover that classifying a d-dimensional (dD) gapped topological phase of generic multibands can reduce to a ($d-1$)D invariant defined on so-called band inversion surfaces (BISs), rendering a bulk-surface duality which simplifies the topological characterization. Further, we show in quenching across phase boundary the (pseudo) spin dynamics to exhibit unique topological patterns on BISs, which are attributed to the post-quench bulk topology and manifest a dynamical bulk-surface correspondence. For this the topological phase is classified by a dynamical topological invariant measured from an emergent dynamical spin-texture field on the BISs. Applications to quenching experiments on feasible models are proposed and studied, demonstrating the new experimental strategies to detect topological phases with high feasibility. This work opens a broad new direction to classify and detect topological phases by non-equilibrium quantum dynamics.

物质的拓扑相现在是凝聚态物理研究的主流，其中拓扑态的分类、综合和检测在近十年来带来了兴奋，但在理论和实验方面仍面临持续挑战。在这里，我们建议建立按整数分类的平衡拓扑量子相的通用非平衡表征，并进一步提出检测此类状态的高精度动力学方案。动力分类理论的框架由基本定理组成。首先，我们发现对通用多带的d维 ( d D) 带隙拓扑相进行分类可以简化为 ($d-\mathrm{1个}$)D 不变量定义在所谓的带反转表面 (BIS) 上，呈现体表面对偶性，从而简化了拓扑表征。此外，我们在跨相界淬火时展示了（伪）自旋动力学以在 BIS 上展示独特的拓扑模式，这归因于淬火后体积拓扑并表现出动态体积表面对应. 为此，拓扑阶段按动态拓扑不变量分类，该不变量从 BIS 上的新兴动态自旋纹理场测量。提出并研究了可行模型在淬火实验中的应用，展示了检测具有高可行性的拓扑相的新实验策略。这项工作开辟了一个广阔的新方向，通过非平衡量子动力​​学对拓扑相进行分类和检测。