标题： 三维时空对称拓扑场论边界理论的单纯形张量网络重正化群 显示英文标题

标题： Simplex tensor network renormalization group for boundary theory of 3+1D symTFT

摘要： 按照arXiv:2210.12127中的构造，我们为3D对称理论开发了一个保持对称性的重整化群（RG）流。 这些理论被表示为一个对称拓扑场理论（symTFT）的边界条件，在我们的情况下，这是一个3+1D的Dijkgraaf-Witten拓扑理论。 边界在几何上被组织成四面体，并被表示为张量网络，我们将其称为“单形张量网络”状态。 每个单形张量被分配了对应于其顶点、边和面的指标。 我们提出了一种数值算法来实现这些边界条件的RG流，并明确展示了其在$\mathbb{Z}_2$对称理论中的应用。 通过在三个拓扑固定点边界之间进行线性插值，我们将由局部和非局部序参数表征的相变映射出来，分别检测0形式和2形式对称性的破缺。 这个形式化方法可以很容易地扩展到其他离散对称群，并且原则上可以推广以描述3D对称拓扑序。 显示英文摘要

摘要： Following the construction in arXiv:2210.12127, we develop a symmetry-preserving renormalization group (RG) flow for 3D symmetric theories. These theories are expressed as boundary conditions of a symTFT, which in our case is a 3+1D Dijkgraaf-Witten topological theory in the bulk. The boundary is geometrically organized into tetrahedra and represented as a tensor network, which we refer to as the "simplex tensor network" state. Each simplex tensor is assigned indices corresponding to its vertices, edges, and faces. We propose a numerical algorithm to implement RG flows for these boundary conditions, and explicitly demonstrate its application to a $\mathbb{Z}_2$ symmetric theory. By linearly interpolating between three topological fixed-point boundaries, we map the phase transitions characterized by local and non-local order parameters, which respectively detects the breaking of a 0-form and a 2-form symmetry. This formalism is readily extendable to other discrete symmetry groups and, in principle, can be generalized to describe 3D symmetric topological orders.