格点上有限对称性的费米子反常
Fermionic Anomalies of Finite Symmetries on Lattices
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- Columbia University（哥伦比亚大学）
- The University of Tokyo（东京大学）
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中文总结 AI 辅助
该研究针对(1+1)和(2+1)维格点有限对称性，开发了费米子't Hooft反常的格点表征方法，发现格点与连续反常存在不匹配，为连续反常的微观格点实现研究提供了动机。
中文摘要 AI 辅助
我们针对(1+1)维和(2+1)维中有限内部对称性的费米子't Hooft反常，开发了一种格点表征方法，该方法以对称短程纠缠（SRE）态的障碍为基础进行构建。我们考虑由格点上的 onsite 费米子和玻色子希尔伯特空间的张量积构成的格点系统，以及由中心扩张$\boldsymbol{\boldsymbol{Z}_2^F\to G_f\to G_b}$给出的有限内部对称性。我们针对给定的对称算子提取了费米子反常指标的层级结构。在(1+1)维中，精确格点对称性由一对上同调数据$\boldsymbol{(n_2,\nu_3)}$表征。对于$\boldsymbol{G_f=G_b\times\boldsymbol{Z}_2^F}$，我们证明，具有平凡反常指标$\boldsymbol{(n_2,\nu_3)}$的对称性是可 onsite 的，因此允许存在对称SRE态，这确立了这些指标能忠实地检测格点反常。与连续QFT相比，我们发现精确格点对称性不会实现连续QFT中额外的$\boldsymbol{H^1(BG_b,\boldsymbol{Z}_2)}$反常层。特别地，对于$\boldsymbol{G_b=\boldsymbol{Z}_2}$，精确格点对称性仅实现连续$\boldsymbol{Z_8}$分类中的偶$\boldsymbol{Z_4}$子群。在(2+1)维中，我们确定了三层连续的上同调数据$\boldsymbol{(n_2,n_3,\nu_4)}$构成的反常层。我们证明，任意一层的非平凡值都会阻碍对称SRE态的存在。对于$\boldsymbol{G_f=G_b\times\boldsymbol{Z}_2^F}$，这还会禁止对称可逆态的存在。当玻色子群$\boldsymbol{G_b}$由费米子宇称进行非平凡扩张时，格点对SRE态的障碍通常与连续't Hooft反常并不一致。我们明确构造了一个(2+1)维中的$\boldsymbol{Z_4^F}$格点对称性，其具有非平凡的格点反常指标，会阻碍对称SRE态的存在，即便它的连续反常是平凡的。我们的结果凸显了格点反常与连续反常之间的不匹配，并为系统研究哪些连续反常允许精确的微观格点实现提供了动机。
英文摘要
We develop a lattice characterization of fermionic 't Hooft anomalies of finite internal symmetries in one and two spatial dimensions, formulated in terms of obstructions to symmetric short-range-entangled (SRE) states. We consider lattice systems formed by tensor product of onsite fermionic and bosonic local degrees of freedom, and finite internal symmetry given by a central extension $\mathbb Z_2^F\to G_f\to G_b$. We extract a hierarchy of fermionic anomaly indices for a given symmetry operator in both (1+1)D and (2+1)D. In (1+1)D, an exact lattice symmetry is characterized by a pair of cohomological data $(n_2,ν_3)$. For $G_f=G_b\times\mathbb Z_2^F$, we further show that a symmetry with trivial lattice anomaly indices $(n_2,ν_3)$ is onsiteable and hence admits a symmetric SRE state, establishing that these indices faithfully detect the lattice anomaly. Comparing with continuum fermionic QFT, we find that exact lattice symmetries do not realize the additional $H^1(BG_b,\mathbb Z_2)$ anomaly layer in continuum QFT. In particular, for $G_b=\mathbb Z_2$, exact lattice symmetries realize only the even $\mathbb Z_4$ subgroup of the continuum $\mathbb Z_8$ classification. In (2+1)D, we identify three successive anomaly layers of cohomological data $(n_2,n_3,ν_4)$. We show that a nontrivial value of any layer obstructs a symmetric SRE state. For $G_f=G_b\times\mathbb Z_2^F$, it also forbids a symmetric invertible state. We also construct a non-onsite $\mathbb Z_4^F$ lattice symmetry in (2+1)D with nontrivial lattice anomaly index that forbids any symmetric invertible states with $c_-\in\mathbb{Z}$, therefore forbids any invertible states compatible with onsite $\mathbb{Z}_4^F$ symmetry. Our results motivate a systematic study of which continuum anomalies admit exact microscopic lattice realizations.
