发表机构
Department of Physics, Harvard University; Department of Physics, Massachusetts Institute of Technology; School of Natural Sciences, Institute for Advanced Study（哈佛大学物理系; 麻省理工学院物理系; 高等研究院自然科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在2+1维量子格点系统中构造了有限2-群对称性的格点实现，涵盖裂2-群和中心2-群，研究了对称性算符、缺陷及规范场化，并提出了可解的对称哈密顿量，计算了基态简并度。
AI 中文摘要
我们在具有有限维张量积希尔伯特空间的${2+1}$维量子格点系统中构造并研究了有限2-群对称性的格点实现。我们聚焦于两类广泛的2-群，其0-形式对称群为$G$，1-形式对称群为$A$：具有平凡Postnikov类${[\beta]\u2208\mathcal{H}^3(G,A)}$的裂2-群，以及具有平凡作用${\rho\colon G\to\text{Aut}(A)}$的中心2-群。在这两种情形下，我们在全张量积希尔伯特空间上构造了对称性算符，当限制到格点$A$ 1-形式对称性的拓扑子空间时，这些算符成为2-群对称性算符。格点裂2-群对称性算符是局域的（onsite），而格点中心2-群对称性算符则不是局域的，只有在引入辅助量子比特（ancillae）后才能变为局域。我们广泛探索了$\rho$和$[\beta]$在这些格点2-群对称性算符中的各种表现，并证明了它们与量子场论预期的吻合。这些表现出现在携带对称荷的算符的变换、格点2-群对称性缺陷的结构，以及通过格点2-群对称性规范场化得到的对偶融合2-范畴对称性中。我们进一步为这两类格点2-群对称性提出了局域对称哈密顿量族，并确定了位于具有自发2-群对称性破缺和非平凡对称性富集拓扑序的相中的精确可解极限。在其中一个极限中，规范场化后的哈密顿量是对应2-群规范理论的精确可解格点实现，我们计算了其基态简并度。
英文摘要
We construct and study lattice realizations of finite 2-group symmetries in ${2+1}$d quantum lattice systems with finite-dimensional tensor-product Hilbert spaces. We focus on two broad classes of 2-groups with 0-form symmetry group $G$ and 1-form symmetry group $A$: split 2-groups with trivial Postnikov class ${[β]\in\mathcal{H}^3(G,A)}$, and central 2-groups with trivial action ${ρ\colon G\to\text{Aut}(A)}$. In both cases, we construct symmetry operators on the full tensor-product Hilbert space that become 2-group symmetry operators when restricted to the topological subspace of the lattice $A$ 1-form symmetry. While the lattice split 2-group symmetry operators are onsite, the lattice central 2-group symmetry operators are not, and can only be made onsite after introducing ancillae. We extensively explore various manifestations of $ρ$ and $[β]$ for these lattice 2-group symmetry operators and demonstrate their agreement with expectations from quantum field theory. These manifestations arise in the transformation of operators carrying symmetry charge, the structure of lattice 2-group symmetry defects, and the dual fusion 2-category symmetries obtained by gauging the lattice 2-group symmetries. We further propose families of local symmetric Hamiltonians for both classes of lattice 2-group symmetries and identify exactly solvable limits lying in phases with spontaneous 2-group symmetry breaking and nontrivial symmetry-enriched topological order. In one such limit, the gauged Hamiltonians are exactly solvable lattice realizations of the corresponding 2-group gauge theories, whose ground-state degeneracies we calculate.
Comments47 pages + appendices
