论文标题
$ \ mathbb {z} _ {n} $ rank-2旋转代码及其低能量有效场理论中的依赖位置的激发和UV/ir混合
Position-Dependent Excitations and UV/IR Mixing in the $\mathbb{Z}_{N}$ Rank-2 Toric Code and its Low-Energy Effective Field Theory
论文作者
论文摘要
我们研究了如何将对称性和拓扑顺序耦合在$ {2+1} $ d $ \ mathbb {z} _ {n} $ n} $ rank-2 toric code for General $ n $中，这是对称级别级别级别的HIGGS阶段的确切可解决的点。存在的对称性富拓扑顺序存在于方形晶格翻译（以及旋转/反射）对称性的非平地实现，在不同的晶格位点上的任何人具有不同的类型，并且属于不同的超选择扇区。我们称这种粒子为“依赖位置的激发”。结果，在排名2的曲奇代码中，Anyons可以在某些方向上通过一个晶格站点跳跃，而仅在其他方向上乘坐$ n $ lattice网站，让人联想到$ {3+1} $ d的Fracton拓扑顺序。我们发现，虽然有$ n^2 $ $ e $费用的口味和$ 2n $ $ m $ fluxes的口味，但没有$ n^{n^{2} + 2n} $ anyon类型。取而代之的是，有$ n^{6} $ ANYON类型，我们可以将Chern-Simons理论与六个$ u（1）$ gauge字段一起描述所有这些理论。虽然晶格翻译置入了任何类型，但我们发现此类排列不能表示为六个$ u（1）$量规字段上的转换。因此，尚不清楚$ u^6（1）$ Chern-Simons理论中翻译对称性的实现。尽管如此，我们还是找到了一种计算理论翻译依赖性特性的方法。特别是，我们发现$ {l_ {x} \ times l_ {y}} $ torus是$ {n^{3} \ gcd（l_ {x}，n），n）\ gcd（l_}，l_ {y}，n），n） $ \ gcd $代表“最大的普通除数”。我们认为，这是紫外/IR混合的表现，它来自晶格对称性和拓扑顺序之间的相互作用。
We investigate how symmetry and topological order are coupled in the ${2+1}$d $\mathbb{Z}_{N}$ rank-2 toric code for general $N$, which is an exactly solvable point in the Higgs phase of a symmetric rank-2 $U(1)$ gauge theory. The symmetry enriched topological order present has a non-trivial realization of square-lattice translation (and rotation/reflection) symmetry, where anyons on different lattice sites have different types and belong to different superselection sectors. We call such particles "position-dependent excitations." As a result, in the rank-2 toric code anyons can hop by one lattice site in some directions while only by $N$ lattice sites in others, reminiscent of fracton topological order in ${3+1}$d. We find that while there are $N^2$ flavors of $e$ charges and $2N$ flavors of $m$ fluxes, there are not $N^{N^{2} + 2N}$ anyon types. Instead, there are $N^{6}$ anyon types, and we can use Chern-Simons theory with six $U(1)$ gauge fields to describe all of them. While the lattice translations permute anyon types, we find that such permutations cannot be expressed as transformations on the six $U(1)$ gauge fields. Thus the realization of translation symmetry in the $U^6(1)$ Chern-Simons theory is not known. Despite this, we find a way to calculate the translation-dependent properties of the theory. In particular, we find that the ground state degeneracy on an ${L_{x}\times L_{y}}$ torus is ${N^{3}\gcd(L_{x},N) \gcd(L_{y},N) \gcd(L_{x},L_{y},N)}$, where $\gcd$ stands for "greatest common divisor." We argue that this is a manifestation of UV/IR mixing which arises from the interplay between lattice symmetries and topological order.
