标题： 量子多体晶格C-R-T对称性：分数量子化、异常和对称质量生成 显示英文标题

标题： Quantum Many-Body Lattice C-R-T Symmetry: Fractionalization, Anomaly, and Symmetric Mass Generation

摘要： 电荷共轭(C)、镜像反射(R)和时间反演(T)对称性，以及内部对称性，对于无质量的马约拉纳费米子和狄拉克费米子是必不可少的。 这些对称性足以排除潜在的费米子双线性质量项，从而确立一个无能隙的自由费米子固定点相位，这是对称质量生成(SMG)转变的关键。 在本工作中，我们通过分析格点量子多体希尔伯特空间中C-R-T-内部对称性群的投影表示（即分数化），系统地研究了所有时空维度中C-R-T-内部对称性的异常。 通过发现与费米子味数相关的C-R-T-内部对称性的异常结构，我们展示了一种推导SMG最小味数的替代方法，该方法与从Kähler-Dirac费米子或边界分类得出的已知结果一致。 我们的发现表明，在一般的空间维度中，要么8个交错马约拉纳费米子的副本，要么4个交错狄拉克费米子的副本可以实现SMG。 通过直接寻找形成满足所有对称性约束的可交换稳定器的四费米子相互作用，我们可以证明显式的SMG闭合保留了代码空间中的唯一基态。 此外，我们建立了交错费米子的对称性算子与自由费米子之间的对应关系，这在促进场论层面的对称性分数化分析中起着关键作用。 显示英文摘要

摘要： Charge conjugation (C), mirror reflection (R), and time reversal (T) symmetries, along with internal symmetries, are essential for massless Majorana and Dirac fermions. These symmetries are sufficient to rule out potential fermion bilinear mass terms, thereby establishing a gapless free fermion fixed point phase, pivotal for symmetric mass generation (SMG) transition. In this work, we systematically study the anomaly of C-R-T-internal symmetry in all spacetime dimensions by analyzing the projective representation (i.e. the fractionalization) of the C-R-T-internal symmetry group in the quantum many-body Hilbert space on the lattice. By discovering the fermion-flavor-number-dependent C-R-T-internal symmetry's anomaly structure, we demonstrate an alternative way to derive the minimal flavor number for SMG, which shows consistency with known results from K\"ahler-Dirac fermion or cobordism classification. Our findings reveal that, in general spatial dimensions, either 8 copies of staggered Majorana fermions or 4 copies of staggered Dirac fermions admit SMG. By directly searching for 4-fermion interactions that form commuting stabilizers respecting all symmetry constraints, we can prove the explicit SMG gapping retained a unique ground state in the codespace. Furthermore, we establish the correspondence between the symmetry operators of staggered fermions and free fermions, which is instrumental in facilitating the analysis of symmetry fractionalization at the field theory level.