标题： 三维费米子拓扑序中强制对称性破缺导致宇称不守恒 显示英文标题

标题： Parity violation as enforced symmetry breaking in 3D fermionic topological order

摘要： 对称性在拓扑相中可以由于固有的不兼容性而内在地被破坏，这一现象在拓扑序的框架下被称为强制对称破缺（ESB）。在我们之前的工作中，我们开发了一个系统性的框架来理解二维可逆拓扑序中的ESB。同时，标准模型中宇称破缺的起源仍然是物理学中最深刻的谜团之一，至今没有明确的解释。在这项研究中，我们通过三维费米子拓扑序（fTO）探讨了宇称对称性的强制对称破缺，为理解宇称破缺的起源提供了潜在的见解。作为最简单的例子，这里我们考虑一个与三维由 $Z_2^f\times Z_2\times Z_8$ 对称性保护的内在相互作用费米子SPT相相关的fTO。我们证明了物理费米子上的时间反演对称性（TRS） ${T}^2=1$ 与此类fTO不兼容；然后，通过所谓的晶体等价原理，我们证明了宇称对称性也与此不兼容。相比之下，具有 ${T}^2={P}_f$ 的常规TRS仍然与此fTO兼容。我们还讨论了一种研究三维fTO中ESB现象的一般框架。 显示英文摘要

摘要： Symmetry can be intrinsically broken in topological phases due to inherent incompatibilities, a phenomenon known as enforced symmetry breaking (ESB) in the framework of topological order. In our previous work, we developed a systematic framework to understand ESB within 2D invertible topological order. Meanwhile, the origin of parity violation in the Standard Model remains one of the most profound mysteries in physics, with no clear explanation to date. In this study, we explore the ESB of parity symmetry by three-dimensional fermionic topological order (fTO), offering potential insights into the origins of parity violation. As the simplest example, here we consider an fTO related to the intrinsic interacting fermionic SPT phase protected by $Z_2^f\times Z_2\times Z_8$ symmetry in three dimensions. We show that time-reversal symmetry (TRS) with ${T}^2=1$ on physical fermions is incompatible with such fTO; then, through the so-called crystalline equivalence principle, we show that the parity symmetry is also incompatible with it. In comparison, conventional TRS with ${T}^2={P}_f$ remains compatible to this fTO. We also discuss a general framework to study the ESB phenomenon for 3D fTO.