论文标题
在$ u（1）$量表理论的限制阶段中，紧急的高对称性受保护的拓扑订单
Emergent higher-symmetry protected topological orders in the confined phase of $U(1)$ gauge theory
论文作者
论文摘要
我们考虑3+1d中的紧凑型$ u^κ（1）$量学理论，$2π$ Quantized拓扑术语$ {\ sum_ {\ sum_ {i，j = 1}^κ\ frac {k_ {k_ {ij}}} {4π} {4π} \ int_} \ int_ {m^4} f^i \ wedge f^j j} $。在量规以下的能量中，距离差距的差距，但该字段理论具有出现的$ {\ Mathbb {z} _ {k_1}^{（1）} \ times \ Mathbb {z} _ {z} _ {k_2}}^（k_2}^（k_2}^（k_2}^（1））$ 1-S-kimny $ 1-1 $ k $和$ \ mathbb {z} _ {0}^{（1）} $的史密斯正常形式的元素被视为$ u（1）^{（1）} $。在$ u^κ（1）$限制阶段中，边界的IR属性由Chern-Simons Field理论描述，并且具有$ {\ Mathbb {Z} _ {k_1}^{（1）} \ times \ times \ times \ MathBb {z} _ {z} _ {k_2} _ {k_2}}^{（1）^{（1）} $ cdots usem be cdots。为了显示这些结果，我们开发了一个玻色晶晶格模型，该模型由该领域理论描述，其属性是其紫外线完成的。上述限制中的晶格模型具有精确的$ {\ mathbb {z} _ {k_1}^{（1）} \ times \ times \ mathbb {z} _ {k_2}}^{（1）} \ times \ cdots \ cdots} $ 1-米米。我们发现，与$ u^κ（1）$量规理论相对应的晶格模型的间隙阶段是对对称的受保护拓扑（SPT）阶段的阶段$ {\ mathbb {z} _ {k_1}^{（1）} \ times \ mathbb {z} _ {k_2}^{（1）} \ times \ cdots} $ 1-米特里，其spt spt spt spt是$ {e^e^e^e^e^e^k _ j j j j j j i^k} b_i \ smile b_j+b_i \ usterset {1} {\ smile} d b_j} e^{iπ\ sum_ {i <j} k_ {ij} \ int d b_i \ int d b_i \ unterset {2}在这里，背景2- chains $ b_i $满足$ {d b_i = \ sum_i b_ {i} k_ {ij} = 0} $ mod $ 1 $并描述了对称性的扭曲$ {\ mathbb {z} _ {k_1}^{（1）} \ times \ mathbb {z} _ {k_2}^{（1）} \ times \ cdots} $ 1-米特里。我们将此一般结果应用于一些简单$ K $矩阵的示例。我们在这些模型的狭窄阶段中发现了非平凡的SPT顺序，并使用相应2组的第四个同胞组讨论了其分类。
We consider compact $U^κ(1)$ gauge theory in 3+1D with the $2π$-quantized topological term ${\sum_{I, J =1}^κ\frac{K_{IJ}}{4π}\int_{M^4}F^I\wedge F^J}$. At energies below the gauge charges' gaps but above the monopoles' gaps, this field theory has an emergent ${\mathbb{Z}_{k_1}^{(1)}\times\mathbb{Z}_{k_2}^{(1)}\times\cdots}$ 1-symmetry, where $k_i$ are the diagonal elements of the Smith normal form of $K$ and $\mathbb{Z}_{0}^{(1)}$ is regarded as $U(1)^{(1)}$. In the $U^κ(1)$ confined phase, the boundary's IR properties are described by Chern-Simons field theory and has a ${\mathbb{Z}_{k_1}^{(1)}\times\mathbb{Z}_{k_2}^{(1)}\times\cdots}$ 1-symmetry that can be anomalous. To show these results, we develop a bosonic lattice model whose IR properties are described by this field theory, thus acting as its UV completion. The lattice model in the aforementioned limit has an exact ${\mathbb{Z}_{k_1}^{(1)}\times\mathbb{Z}_{k_2}^{(1)}\times\cdots}$ 1-symmetry. We find that a gapped phase of the lattice model, corresponding to the confined phase of the $U^κ(1)$ gauge theory, is a symmetry protected topological (SPT) phase for the ${\mathbb{Z}_{k_1}^{(1)}\times\mathbb{Z}_{k_2}^{(1)}\times\cdots}$ 1-symmetry, whose SPT invariant is ${e^{iπ\sum_{I, J}K_{IJ}\int B_I\smile B_J+B_I\underset{1}{\smile} d B_J}e^{iπ\sum_{I< J}K_{IJ}\int d B_I\underset{2}{\smile}d B_J}}$. Here, the background 2-cochains $B_I$ satisfy ${d B_I=\sum_I B_{I}K_{IJ} = 0}$ mod $1$ and describe the symmetry twist of the ${\mathbb{Z}_{k_1}^{(1)}\times\mathbb{Z}_{k_2}^{(1)}\times\cdots}$ 1-symmetry. We apply this general result to a few examples with simple $K$ matrices. We find the non-trivial SPT order in the confined phases of these models and discuss its classifications using the fourth cohomology group of the corresponding 2-group.
