标题： 克莱因瓶上的二维对称保护拓扑相之间的转变 显示英文标题

标题： Transition between 2D Symmetry Protected Topological Phases on a Klein Bottle

摘要： 具有非平凡拓扑的流形在研究物质的拓扑相中起着关键作用。 在本文中，我们研究了当系统置于非可定向流形——克莱因瓶上时，$2+1$D$Z_2$对称性保护拓扑（SPT）相的非平凡对称性响应。 特别是，我们发现当在克莱因瓶的方向保留循环上插入对称性缺陷时，系统的基态会获得额外的电荷。 这种响应在进入平凡SPT相的转变点处仍然定义良好，导致基态出现精确的二重简并，与系统大小无关。 我们使用SPT相的精确可解格点模型以及转变过程中的数值工作来演示这种对称性响应。 我们探讨了这一结果与$3+1$D$Z_2$规范理论的模变换以及$Z_2$SPT相中宇称对称性的涌现性质之间的联系。 显示英文摘要

摘要： Manifolds with nontrivial topology play an essential role in the study of topological phases of matter. In this paper, we study the nontrivial symmetry response of the $2+1$D $Z_2$ symmetry-protected topological (SPT) phase when the system is put on a non-orientable manifold -- the Klein bottle. In particular, we find that when a symmetry defect is inserted along the orientation-reserving cycle of the Klein bottle, the ground state of the system gets an extra charge. This response remains well defined at transition points into the trivial SPT phase, resulting in an exact two-fold degeneracy in the ground state independent of the system size. We demonstrate the symmetry response using exactly solvable lattice models of the SPT phase, as well as numerical work across the transition. We explore the connection of this result to the modular transformation of the $3+1$D $Z_2$ gauge theory and the emergent nature of the parity symmetry in the $Z_2$ SPT phase.