论文标题
具有扭曲边界条件及其对称性的分形剂
Fractons with Twisted Boundary Conditions and Their Symmetries
论文作者
论文摘要
我们在平坦的三道弯曲上研究了几个外来系统，包括X-Cube型号，并在$ xy $平面中扭曲。基础状态变性事实证明是各种几何参数的敏感功能。从晶格开始，具体取决于我们如何达到连续限制，我们发现基态变性的不同值。然而，自然的连续体极限，该变性的定义明确（尽管是无限）的。我们还发现了令人惊讶的全球对称性，价格为$ 2+1 $和$ 3+1 $尺寸系统。它起源于基础子系统对称性，但实现的方式取决于扭曲。特别是，在首选的坐标框架中，扭曲的两道torus $τ=τ_1 +iτ_2$的模块化参数具有有理$τ_1= k / m $。然后，在基于$ u（1）\ times u（1）$子系统对称的系统中，例如动量和绕组对称性或电动和磁性对称性，新对称性是一个项目实现的$ \ mathbb {z} _m \ times \ times \ mathbb {z} _m $ $ $ $ - $ m $ m-m-$ -folder，在基于$ \ mathbb {z} _n $ symmetries的系统中，就像X-Cube模型一样，这两个$ \ Mathbb {Z} _M $因子中的每一个都被$ \ Mathbb {Z} _ {\ gcd（n，m）} $替换。
We study several exotic systems, including the X-cube model, on a flat three-torus with a twist in the $xy$-plane. The ground state degeneracy turns out to be a sensitive function of various geometrical parameters. Starting from a lattice, depending on how we take the continuum limit, we find different values of the ground state degeneracy. Yet, there is a natural continuum limit with a well-defined (though infinite) value of that degeneracy. We also uncover a surprising global symmetry in $2+1$ and $3+1$ dimensional systems. It originates from the underlying subsystem symmetry, but the way it is realized depends on the twist. In particular, in a preferred coordinate frame, the modular parameter of the twisted two-torus $τ= τ_1 + i τ_2$ has rational $τ_1 = k / m$. Then, in systems based on $U(1)\times U(1)$ subsystem symmetries, such as momentum and winding symmetries or electric and magnetic symmetries, the new symmetry is a projectively realized $\mathbb{Z}_m\times \mathbb{Z}_m$, which leads to an $m$-fold ground state degeneracy. In systems based on $\mathbb{Z}_N$ symmetries, like the X-cube model, each of these two $\mathbb{Z}_m$ factors is replaced by $\mathbb{Z}_{\gcd(N,m)}$.
